### MathStat

```Statistical Inference and Regression
Analysis:
Stat-GB.3302.30, Stat-UB.0015.01
Professor William Greene
Stern School of Business
IOMS Department
Department of Economics
Part 2 – A
Expectations of
Random Variables
2-A Expectations of Random Variables
2-B Covariance and Correlation
2-C Limit Results for Sums
Part 2 – Expectations of Random Variables
3/124
Expected Value of a
Random Variable
Weighted average of the values taken
by the variable
Discrete
E[X]   all values taken by X x Pr ob(X  x)
Continuous
E[X]  

xf (x)dx

(Density equals zero outside the range of x.)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
4/124
Discrete Uniform
X = 1,2,…,J
 Prob(X = x) = 1/J
 E[X] = 1/J + 2/J + … + J/J
= J(J+1)/2 * 1/J
= (J+1)/2
 Expected toss of a die = 3.5 (J=6)
 Expected sum of two dice = 7. Proof?

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
5/124
Poisson ()
e  x
E[X]   x 0 x
x!

e  x
  x 1 x
(drop zero term)
x!

e  x 1 
x
1 
   x 1

 factor out  and

(x  1)! 
x! (x  1)! 

e  z
=   z 0
(let z = x-1; z goes from 0 to )
z!
=  (probabilities sum to 1)

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
6/124
Poisson (5)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
7/124
The St. Petersburg Paradox
Coin toss game. If first heads comes up on
nth toss, you win \$2n
 Entry fee to play a game is \$C
 Expected value of the game = E[Win]
-C + (½)21 + (½)222 + … + (½)k2k  
Game has infinite value. Noone would pay
very much to play. Why not?

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
8/124
Continuous Random Variable
E[X]  


xf (x)dx, 'support of x' = {x : f(x) > 0}
1
Continuous uniform: f(x) =
I(x  [a, b])
ba
b
1
1  x2 b 
E[x]   x
dx 

a 
a
ba
ba  2 
b 2  a 2 (b  a)(b  a) b  a



(the midpoint)
2(b  a)
2(b  a)
2
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
9/124
Gamma Random Variable
 P e x x P 1
f (x) 
, x  0,   0, P  0
(P)
Gamma function: (P)= 

e  t t P 1dt.
0
Results: (P)  (P  1)(P  1) = (P-1)! (Show by integration by parts)
(1/2) =  (We will prove this later)
Implication: We know
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
(P)
. Expected value is
P

 P  x P
 P (P  1) P
x P 1
xe x dx 
e x dx 

P 1

0
(P)
(P) 

e x x P 1dx 
P 
(P) 0
Mushroom and Onion
9.2%
f (x)dx  1.
Frequency
0
0
Listing


Percent
So,


20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
10/124
Gamma Function: (1/2)=
(1/ 2)  

e t dt  
t
0
1 1
2

 t  12
e t dt
0
Change of variable from t to z= t so t = z 2 and dt=2zdz
(1/ 2)  

e
0
 z2
z 2zdz  2 
1

0
 z2
e dz
Change of variable from x to z 2 so z=x/ 2 and dz=1/ 2dx
(1/ 2)  2 

e
0
 x 2 /2
 1    x 2 /2 
2
1/ 2dx 
2 
dx 
 0 e
2
 2 

 1    x 2 /2  1
2
1
e
dx

so

(1/
2)

2



  
 0
2
2
 2 
 2
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
11/124
Expected Value of a Linear
Translation
Z = aX+b
 E[Z] = aE[X] + b
 Proof is trivial using the definition of
the expected value and the fact that
the density integrates to 1 to
have E[b]=b.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
12/124
Normal(,) Variable

From the definition of the random
variable,  is the mean.

Proof in Rice (119) uses the linear
translation.
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Frequency
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
If X ~ N[0,1], X +  ~ N(,)
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
13/124
Cauchy Random Variables
f(x)=(1/) 1/(1+x2)
 Mean does not exist. No higher moments
exist.
 If X~N[0,1] and Y ~ N[0,1] then X/Y has the
Cauchy distribution.
 Many applications obtain estimates of
interesting quantities as ratios of estimators
that are normally distributed.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
14/124
Cauchy Random Sample
1016
F req u en cy
762
508
254
0
- 562. 137
- 410. 638
- 259. 138
- 107. 639
43. 860
195. 359
346. 858
498. 357
Z
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
15/124
Expected Value of a
Function of X
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Percent

Frequency

Listing

Y=g(X)
One to one case
 E[Y] = expected value of Y(X) – find the
distribution of the new variable
 E[g(X)] = x g(x)f(x) will equal E[Y]
Many to one case – similar argument. Proceed
without the transformation of the random
variable.
E[g(X)] is generally not equal to g(E[X]) if
g(X) is not linear
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
16/124
Linear Translation
Z = aX+b
 E[Z] = E[aX+b]
 E[Z] = aE[X] + b
 Proof is trivial using the definition of
the expected value and the fact that
the density integrates to 1 to E[b]=b.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
17/124
Powers of x - Moments

Moment = E[Xk] for positive integer x
Raw moment: E[Xk]
 Central moment: E[(X – E[X])k]


Standard notation
E[Xk] = k
 E[(X – E[X])k] = k
 Mean = 1 = 

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
18/124
Variance as a g(X)


Variance = E[(X – E[X])2]
Standard deviation = square root of
variance is usually more interesting
Discrete
Var[X] =
2
(x


)
Pr ob(X  x)
x
Continuous
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%

Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

(x  ) 2 f (x)dx
Frequency
Var[X] =

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
19/124
Variance of a Translation:
Y = a + bX
Plain
32.5%
800000
800000
500000
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
900000
Percent
Scatterplot of Listing vs IncomePC
900000
400000
Mushroom
16.2%
Standard deviation of Y = |b|S.D.(X)
Frequency
Sausage
5.8%

Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
Var[bX] = b2Var[X]
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
Var[a] = 0
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
20/124
Shortcut

Var[X] = E[X2] - {E[X]}2
Uniform (0,1)
1
E[X] 
2
E[X ]  
2
3
x
x 21dx 
3
1
0
1
0
1

3
2
1 1
1
Var[X]     
3  2  12
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
21/124
Bernoulli
Prob(X=1)=; Prob(X=0)=1- 
 E[X] = 0(1- ) + 1 = 
 E[X2] = 02(1- ) + 12 = 
 Var[X] =  - 2 = (1- )

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
22/124
Poisson: Factorial Moment
e  x

e  x
x(x  1)
  x 2 x(x  1)
x!
x!
e  x
.
(x  2)!
E[X  X]  E[X(X  1)]   x 0

2
= x 2

 z
e  z  2

e

Now let z = x-2, sum is  z 0
  2  z 0
= 2
z!
z!
E[X 2 ]  E[X]   2 so E[X 2 ]   2  

Var[X]  E[X 2 ]  {E[X]}2   2     2  
Variance of Poisson variable equals the mean
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
23/124
Normal Moments
x = Normal[,] = N[0,1] + 
2

1
1 x -μ 
f(x) =
exp - 
 , -  < x < + 
σ 2π
 2  σ  
  Mean
 = Standard deviation
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
24/124
Gamma Random Variable
 P e x x P 1
f (x) 
, x  0,   0, P  0
(P)
Expected value is
P 
 P  x P
 P (P  1) P
x P 1
xe x dx 
e x dx 

P 1


0
0
(P)
(P)
(P) 

Expected square is
 P  2 x P 1
 P  x P 1
 P (P  2)
x e x dx 
e x dx 


0
0
(P)
(P)
(P)  P  2
(P  1)(P  1) (P  1)P(P) P(P  1)



.
2
2
2
(P)
(P)

P(P  1)  P 
P
Variance 
   2
2
 
2
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
25/124
Chi Squared [1]
Plain
32.5%
800000
800000
500000
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
900000
= 1
Percent
Scatterplot of Listing vs IncomePC
900000
400000
Mushroom
16.2%
Variance = P/ 2 = (½)/[(½)2] = 2
= P/  = (½)/(½)
Frequency
Sausage
5.8%

Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
Mean
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
Chi squared [1] = Gamma(½, ½)
P = ½, =½
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
26/124
Higher Moments

Skewness: 3.
0 for all symmetric distributions (not
just the normal)
 Standardized measure 3/3


Kurtosis: 4.
Standardized 4/4.
 Compare to normal, 3
 Degree of excess = 4/4 – 3.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
27/124
Symmetric and Skewed
Distributions
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
28/124
Kurtosis: t[5] vs. Normal
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
= 3, Excess = 0
= 6/(k-4); for t[5] = 6/(5-4) = 6.
0
1000000
60
800000
40
Listing
Kurtosis of normal(0,1)
Excess Kurtosis of t[k]
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
29/124
Approximations for g(X)

g(X) = continuous function
g() exists
Continuous first derivative not equal to zero at 


Taylor series approximation around mu
 g(X) = g() +
g’()(X- )
+ ½ g’’()(X- )2
(+ higher order terms)

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
30/124
Approximation to the Mean
g(X) ~ g() + g’()(X- )
+ ½ g’’()(X - )2
 E[g(X)] ~ E[approximation]
= g() + 0 + ½ g’’() E[(X - )2]
= g() + ½ g’’() 2

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
31/124
N[, ].
g(X)=exp(X).
= exp( +  2/2).
= exp() + ½ exp()  2
True mean
Approximation:
Example:  =0, s = 1,
True mean
= exp(.5)
Approximation
= exp(0) + .5*exp(0)*1
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
= 1.6487
= 1.5000
0
1000000
60
800000
40
Listing
Example:
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
32/124
Delta method: Var[g(X)]
Use linear approximation
 g(X) ~ g() + g’()(X - )
 Var[g(X)] ~ Var[approximation]
= [g’()]22
 Example: Var[X2] ~ (2)22

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
33/124
Delta Method – x ~ N[, 2]


Approximate


Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
N[0,1], exact mean and variance are exp(.5)
=1.648 and exp(1)(exp(1)-1) = 4.671.
Approximations are 1.5 and 1 (!)
Percent

= exp() + ½ exp() 2
= [exp()]2 2
E*[y]
V*[y]
Frequency

= exp( + ½ 2)
= exp(2 + 2)[exp(2) – 1]
E[y]
Var[y]
Listing

y = g(x) = exp(x) ~ lognormal
Exact
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
34/124
Moment Generating Function


Let g(X) = exp(tX)
M(t) = E[exp(tX)] = the moment generating
function for random variable X.
M(t)   x e p ( x) or

tx

etx f ( x)dx

If M(t) exists in a neighborhood of zero then
M(t) <== > p(x) or f(x)
One to one correspondence between probability
distribution and moment generating function.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
35/124
MGF Bernoulli
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Percent
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Frequency
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
E[exp(tX)] = (1- )exp(0t) + exp(1t)
= (1 - ) + exp(t).
Listing

P(x) = (1-) for x=0 and  for x=1
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
36/124
MGF Poisson
tx 
e e 
x!
M(t)   x 0

e
  x 0

800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
x!
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
x 0
 exp[(e  1)]
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%


x
t
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
e

t
ax
 ea
x!
x 0
M(t) = e e
 e 
x
x!

-λ λe t
 e 
t
Frequency

Result

x
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
37/124
MGF Gamma
E[exp(tx)]  

0
P x P 1

e x
tx
e
dx
( P )


 (  t ) x P 1

e
x dx

( P ) 0
P

( P )
  



P
 ( P ) (  t )
t 
P
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
P
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
38/124
MGF Normal
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
This is the moment generating
function for N[,2]
Frequency

MY(t) for Y = X +  is
exp(t)MX(t) = exp[t + ½ 2t2]
Listing

MX(t) for X ~ N[0,1] is exp(½ t2)
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
39/124
Generating the Moments
rth derivative of M(t) evaluated at t
= 0 gives the rth raw moment, r’
M(r)(t) = drM(t)/dtr |t=0
= equals rth raw moment.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
40/124
Poisson MGF
= exp((exp(t) – 1)); M(0)=1
 M’(t) = M(t) * exp(t);
M’(0)= 

 = M’(0)=1  1 = 
 2’ = E[X2] = M’’(0) = M’(0) exp(0)
+ exp(0)M(0)
= 2 + 
 Variance = 2’ - 2 = 
 M(t)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
41/124
Useful Properties

MGF of X = MX(t) and y = a+bX then

MY(t) for y is exp(at)MX(bt)

For independent X and Y,
MX+Y (t) = is MX(t)MY(t)

The sequence of moments does not uniquely
define the distribution
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
42/124
Side Results
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

MGF MX(t) = E[exp(tx)] does not always exist.
Characteristic function E[exp(itx)] always exists.
Used to prove central limit theorems
Cumulant generating function logMX(t)
is sometimes useful. Cumulants are functions
of moments. First cumulant is the mean,
second is the variance.
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – B
Covariance and
Correlation
Part 2 – Expectations of Random Variables
44/124
Covariance
Random variables X,Y with joint
discrete distribution p(X,Y) or
continuous density f(x,y).
 Covariance = E({X – E[X]}{Y-E[Y]})
= E[XY] – E[X] E[Y].
 (Note, Covariance of X,X = Var[X].
 Connection to joint distribution and
covariation

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
45/124
Correlation and Covariance
Cov(X,Y)
Var[X]Var[Y]
Correlation Coefficient =  =
By Cauchy - Schwarz inequality, -1    1.
  1 if and only if Y = a + bX with b > 0
  1 if and only if Y = a + bX with b < 0
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
46/124
Correlated Populations
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
47/124
Correlated Variables


X1 and X2 are independent with means 0 and
standard deviations 1.
Y = aX1 + bX2. Choose a and b such that



X1 and Y have means 0, standard deviation 1
and correlation rho.
Var[Y]
= a 2 + b2 = 1
Cov[X1,Y] = a = .
b = sqr(1 – 2)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
48/124
Conditional Distributions
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Percent

Frequency

Listing

f(y|x) = f(y,x) / f(x)
Conditional distribution of y given a
realization of x
Conditional mean = mean of the
conditional random variable =
regression function
Conditional variance = variance of
conditional random variable =
scedastic function
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
49/124
Litigation Risk Analysis
Form probability tree for decisions and outcomes
Determine conditional expected payoffs (gains or
losses)
Choose strategy to optimize expected value of payoff
function (minimize loss or maximize (net) gain.
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
50/124
Litigation Risk Analysis: Using
Probabilities to Determine a Strategy
P(Upper path) = P(Causation|Liability,Document)P(Liability|Document)P(Document)
= P(Causation,Liability|Document)P(Document)
= P(Causation,Liability,Document)
= .7(.6)(.4)=.168. (Similarly for lower path, probability = .5(.3)(.6) = .09.)
Two paths to a favorable outcome. Probability =
(upper) .7(.6)(.4) + (lower) .5(.3)(.6) = .168 + .09 = .258.
How can I use this to decide whether to litigate or not?
800000
800000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
Frequency
Boxplot of Listing
Listing
Pepper and Onion
7.3%
Suppose the cost to litigate = \$1,000,000 and a favorable outcome pays \$3,000,000.
What should you do?
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Percent
Meatball
Garlic 5.0%
2.3%
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
51/124
Joint Normal Random
Variables
2
2


 x  x
1
1  x   x   y   y 

f(x,y)=
exp
  2 
  
2 
2

2(1


)


x
2x  y 1  

x

y






  Correlation of X and Y.
x  y  Covariance of X and Y
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

  y  y 

 
 



y


20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
52/124
Conditional Normal
2
 


y

[



(

/

)(x


)]
1
1 
y
y
x
x
 
f (y | x) 
exp  
2
2 


 y 2 1   2

1


y
 
 
  y   2 
1
1 

y|x


exp
 

 2   y|x  
 y|x 2

 
Conditional Mean Function = E[y|x] =  y  ( y /  x )(x   x )     x
Conditional Variance Function = Var[y|x] = 2y (1  2 ) (not a function of x)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
53/124
Y and
Y|X
Y
X
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
X
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
54/124
Application: Conditional Expected
Profits and Risk
You must decide how many copies of your self published novel to print . Based on market
research, you believe the following distribution describes X, your likely sales (demand).
x
P(X=x)
25
.10
(Note: Sales are in thousands. Convert your final result to
40
.30
dollars after all computations are done by multiplying your
55
.45
final results by \$1,000.)
70
.15
Printing costs are \$1.25 per book. (It’s a small book.) The selling price will be \$3.25. Any
unsold books that you print must be discarded (at a loss of \$1.25/copy). You must decide how
many copies of the book to print, 25, 40, 55 or 70. (You are committed to one of these four – 0
is not an option.)
A. What is the expected number of copies demanded.
B. What is the standard deviation of the number of copies demanded.
C. Which of the four print runs shown maximizes your expected profit? Compute all four.
D. Which of the four print runs is least risky – i.e., minimizes the standard deviation of the
profit (given the number printed). Compute all four.
E. Based on C. and D., which of the four print runs seems best for you?
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
55/124
X = Sales (Demand)
x
P(X=x)
25,000
.10
40,000
.30
55,000
.45
70,000
.15

A. Expected Value =
x  P(X=x)
all values of x
= .1(25,000) + .3(40,000) + .45(55,000) + .15(70,000)
= 49,750
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
56/124
B. Standard Deviation
Get the Variance First
2 

(x - E[x]) 2  P(X=x)
all values of x
= .1(25,000 - 49,750) 2  .3(40,000 - 49,750) 2
+ .45(55,000 - 49,750) 2 + .15(70,000 - 49,750) 2
= 163,687,500
Standard Deviation = square root of variance.

=
163,687,500 = 12,794.041
There is a shortcut
2    all values of x x 2  P(X=x)    2


2 

(x - E[x]) 2  P(X=x)
all values of x
.1(25,0002 )  .3(40,000 2 ) + .45(55,0002 ) + .15(70,0002 )  - 49,750 2
= 163,687,500
=
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
57/124
x
P(X=x) Revenue per book = \$3.25
25,000 .10
Cost per book
= \$1.25
40,000 .30
Profit per book sold = \$2.00/book
55,000 .45
70,000 .15
Expected Profit | Print Run = 25,000 is \$2  25,000 = \$50,000
(Demand is guaranteed to be at least 25,000)
Expected Profit | Print Run = 40,000 is \$2  .9  40,000
+ .1  (\$2  25,000 - \$1.25  15,000) = \$75,125
(If print 40,000, .9 chance sell all and .1 chance sell only 25,000)
Expected Profit | Print Run = 55,000 is \$2  .6  55,000
+ .1  (\$2  25,000 - \$1.25  30,000)
+ .3  (\$2  40, 000  \$1.25 15, 000) = \$85,625
Expected Profit | Print Run=70,000 is \$2  .15  70,000
+ .1  (\$2  25,000 - \$1.25  45,000)
+ .3  (\$2  40, 000  \$1.25  30000)
+ .45  (\$2  55, 000  \$1.25  15000) = \$74,187,50
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
58/124
Expected Profit Given Print Run
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
59/124
Variances
Print Run = 25,000. Variance = 0. Std. Dev. = 0 Demand will be at least 25,000.
Print Run = 40,000. Variance =
.1*[(2* 25000  1.25*15000)  75,125]2 
(if demand is only 25,000)
 75,125)]2
.9*[(2* 40000)
(if demand is  40,000)
Standard Deviation = square root = \$14625
Print Run = 55,000. Variance =
.1*[(2* 25000  1.25*30, 000)  85, 625]2 
.3*[(2* 40000)  1.25*15, 000)  85, 625] +
.6*[(2*55, 000
 85,625]
(if demand is only 25,000)
(if demand is  40,000)
(if demand is  55,000)
Standard Deviation = square root = \$32,702.49
Print Run = 70,000. Variance =
.1* [(2* 25000  1.25* 45000)  74,187.5]2 
(if demand is only 25,000)
.3* [(2* 40000  1.25*30, 000)  74,187.5]2 +
(if demand is  40,000)
.45*[(2*55, 000  1.25*15, 000)  74,187.5]2 +
(if demand is  55,000)
 74,187.5]2
.15*[2*70, 000
(if demand is  70,000)
Standard Deviation = square root = \$41,580.64
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
60/124
Run=70,000
Run=55,000
Run=40,000
Run=25,000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
61/124
Run=70,000
Run=55,000
Run=40,000
Run=25,000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
62/124
Run=70,000
Run=55,000
Run=40,000
Run=25,000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
63/124
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
64/124
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
65/124
Useful Theorems - 1
E[Y] = EX[EY[Y|X]]
 Expectation over X of EY[Y|X]
 Law of Iterated Expectations

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
66/124
Example: Hierarchical Model
e  (x ) (x) y
p( y | x) 
, y  0,1,...,   0
y!
f ( x)  e x , x  0,   0
e  ( ) x (x) y
f ( y, x)  p( y | x) f ( x) 
y  0,1,..., x  0, ,   0
y!
e  ( ) x (x) y
E[ y ]   y  0 yp( y )   y  0 y 
dx
0
y!
But, Y|X is Poisson with parameter (x) so E[y|x] = x


E[Y] = E[E[Y|X]] = E[x] = E[X] =
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

.

0
1000000
60
800000
40
Listing

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
67/124
Useful Theorems - 2
Decomposition of variance
 Var[Y] = Var[E[Y|X]] + E[Var[Y|X]]

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
68/124
Bivariate Normal
E[Y|x]  (Y   X )  x
Var[Y|x]  Y2 (1  2 )
E[Var[Y|x]]  E[Y2 (1  2 )]  Y2 (1   2 )
Var[E[y|x]=Var[x]  2  2X
 XY Y2  XY
 = 2  2 2 (multiply and divide by Y2 .)
X
 X Y
2
2
2
2 2 2








2 2
2 2
Y XY Y XY X
XY XY
Y Y  X
 X 



Y
2
2 2
2
2
2
2 2
 X Y  X Y
 X Y Y  X
Var[Y]  Y2 (1  2 )  2 Y2  Y2
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
69/124
Useful Theorems - 3

Cov(X,Y)=Cov(X,E[Y|X])

In the hierarchical model, E[y|x]=x so
Cov(X,Y)=Cov(X, X)= Var[X]= /2
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
70/124
Mean Squared Error
Error of c as a predictor of Y is (Y - c)
Expected squared error is EY [(Y-c)2 ]
EY [{(Y  )  (  c)}2 ]  EY [(Y  ) 2 ]  EY (  c) 2 ]  2E Y [(Y  )(  c)]
(  c) is not a random variable so the third term is zero, leaving
EY [(Y-c) 2 ]  Var[Y]  (  c) 2
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
71/124
Minimum MSE Predictor
Error of h(X) as a predictor of Y = (Y - h(X))
Expected squared error for a given X is EY [(Y-h(X)) 2 | X]
Expected squared error over X is E X EY [(Y-h(X)) 2 | X]
Add and subtract E[Y|X] = Y | X
E X EY [{(Y-Y | X )+(Y | X -h(X))}2 | X ]
 E X EY [(Y  Y | X )2 | X ]  E X EY (Y | X  h(X)) 2 | X ]
 2E X EY [(Y  Y | X )(Y | X  h( X ))]
The first term is E X [Var[Y|X]].
The second is E X [Y | X  h(X)) 2 | X ]  0 if h(X) = Y | X
E X EY [Y  Y | X ] = 0, so the third term is zero.
This implies that the minimum mean squared predictor is the
conditional mean function.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
72/124
Variance of the Sum of X and Y
Var[X+Y] = EYEX[ {(X+Y) - (X + Y) }2 ]
= EYEX[ {(X- X) + (Y- Y)}2]
= EYEX[ (X - X)2] + EYEX[(Y- Y )2]
+ 2 EYEX[(X- X)(Y- Y)]
= EX[ (X - X)2] + EY[(Y- Y )2]
+ 2 EYEX[(X- X)(Y- Y)]
= Var[X] + Var[Y] + 2Cov(X,Y)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
73/124
Variance of Weighted Sum
Var[aX+bY] = Var[aX] + Var[bY] +2Cov(aX,bY)
= a2Var[X] + b2Var[Y]
+ 2ab Cov(X,Y).
Also, Cov(X,Y) is the numerator in ρxy, so
Cov(X,Y) = ρxy σx σy.
ax by  a   b   2abxy x y
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
2
y
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
2
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
2
x
Frequency
2
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
74/124
Application - Portfolio
You have \$1000 to allocate between assets A and B.
The yearly returns on the two assets are random
variables rA and rB.
The means of the two returns are
E[rA] = μA and E[rB] = μB
The standard deviations (risks) of the returns are σA and
σB.
The correlation of the two returns is ρAB
You will allocate a proportion w of your \$1000 to A and
(1-w) to B.
Pie Chart of Percent vs Type
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
Mushroom and Onion
9.2%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing

Percent

Frequency

Listing

Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
75/124
Risk and Return
Your expected return on each dollar is
E[wrA + (1-w)rB] = wμA + (1-w)μB
 The variance your return on each dollar is
Var[wrA + (1-w)rB]
= w2 σA2 + (1-w)2σB2 + 2w(1-w)ρABσAσB
 The standard deviation is the square root.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
76/124
Risk and Return: Example
Suppose you know μA, μB, ρAB, σA, and σB (You have watched
these stocks for many years.)
 The mean and standard deviation are then just functions of w.
 I will then compute the mean and standard deviation for different
values of w.
 For example, μA = .04, μB, = .07
σA = .02, σB,=.06, ρAB = -.4
E[return] = w(.04) + (1-w)(.07)
= .07 - .03w
SD[return] = sqr[w2(.022)+ (1-w)2(.062) + 2w(1-w)(-.4)(.02)(.06)]
= sqr[.0004w2 + .0036(1-w)2 - .00096w(1-w)]

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
77/124
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
78/124
Mean and Variance of a Sum
Random Variables x1 , x 2 , ..., x n
1 , 1 , ...,  n
Means:
Variances and Covariances ij , i=1,...,n and j=1,...,n
Sum: x1 + x 2 + ... + x n
Mean: 1 +  2 + ... +  n
E[{(x1 -1 )  ...  (x n - n )}2 ]
 
 
=
=
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
j 1
n
n
i 1
j 1
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
4
5
200000
2
1
100000
15000
0
200000
400000
600000
Listing
800000
1000000
Mean
StDev
N
369687
156865
51
80
8
300000
10
Normal
10
500000
400000
20
300000
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
100
12
700000
600000
70
ij
j 1
14
800000
80
400000
100000
15000
369687
156865
51
0.994
0.012

n
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
500000
200000
Cov(x i ,x j )   i 1
Probability Plot of Listing
600000
200000
E(x i -i )(x j - j )
n
99
700000
300000
100000
i 1
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
n
(x i -i )(x j - j ) 

j 1
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
n
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

n
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
n
Percent
= E   i 1

Frequency
Variance:
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
79/124
Extension: Weighted Sum
Random Variables x1 , x 2 , ..., x n
1 , 1 , ...,  n
Means:
Variances and Covariances ij , i=1,...,n and j=1,...,n
Weighted Sum: w1x1 + w 2 x 2 + ... + w n x n
Mean:  i1 w i i
n
E[{w1 (x1 -1 )  ...  w n (x n - n )}2 ]
 
 
=
=
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
n
n
i 1
j 1
w i w j Cov(x i ,x j )   i 1
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
4
5
200000
2
1
100000
15000
0
200000
400000
600000
Listing
800000
1000000
Mean
StDev
N
369687
156865
51
80
8
300000
10
Marginal Plot of Listing vs IncomePC
Normal
100
10
500000
400000
20
300000
100000
15000
60
50
40
w i w j ij
12
700000
600000
70
j 1
Empirical CDF of Listing
14
800000
80
400000
200000
369687
156865
51
0.994
0.012
n
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
500000

n
Probability Plot of Listing
600000
200000
 w i (x i -i )   w j (x j - j ) 
E  w i w j (x i -i )(x j - j ) 
99
700000
300000
100000
j 1
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
i 1
j 1
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
n
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
n
n
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing

n
Percent
= E   i 1

Frequency
Variance:
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
80/124
More General Portfolio Problem
Assets A1 , A 2 , ..., A n .
Expected Returns 1 ,..., n .
Each is random with variance ii
Covariance of return on asset i and asset j is ij , ii  i2
Portfolio is a set of weights w1 w 2 w 3 ... w n that sum to 1.
Variance of return is V =
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
30
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
Mean
StDev
N
369687
156865
51
80
8
4
200000
Normal
10
500000
300000
0
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
100
12
700000
400000
10
17500
Histogram of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
wi w j ij
900000
Mean
StDev
N
P-Value
95
600000
i 1
Scatterplot of Listing vs IncomePC
Normal - 95% CI
99
700000
300000
100000
Probability Plot of Listing
n
wi i
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
i 1
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
n
0
1000000
60
800000
40
Listing


Expected return is M =
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
81/124
Optimal Portfolio?
Minimize the risk while obtaining a specific expected return

= 
n
Minimize V =
Subject to
M*
i 1
wi w j ij
n
i 1

wi i and
n
i 1

wi  1
Choose the set (vector) of weights to minimize V subject to
achieving a specified expected return. (Mathematical programming
problem.) Alternatively, maximize expected return subject to a
specified risk
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
100000
15000
369687
156865
51
0.994
0.012
60
50
40
20
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
4
5
200000
2
1
100000
15000
0
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
300000
10
Mean
StDev
N
10
500000
400000
30
Normal
100
12
700000
600000
70
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
80
300000
200000
100000
90
400000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
500000

wi  1
Scatterplot of Listing vs IncomePC
Normal - 95% CI
600000
200000
i 1
Probability Plot of Listing
99
700000
300000
n
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball

wi w j ij and
Percent
Pepperoni
21.8%
i 1
Listing
Meatball
Garlic 5.0%
2.3%
n
V =
wi i
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
i 1
Percent
Subject to
n
Frequency


Maximize M* =
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
82/124
Sums of Independent Variables
Pie Chart of Percent vs Type
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
Mushroom and Onion
9.2%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing

Percent

Frequency

Listing

Suppose P is sales of a store. The accounting period
starts with total sales = 0
On any given day, sales are random, normally distributed
with mean μ and standard deviation σ. For example, mean
\$100,000 with standard deviation \$10,000
Sales on any given day, day t, are denoted Δt
 Δ1 = sales on day 1,
 Δ2 = sales on day 2,
Total sales after T days will be Δ1+ Δ2+…+ ΔT
Therefore, each Δt is the change in the total that occurs on
day t.
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
83/124
Behavior of the Total
P1 = Δ1
P2 = Δ1 + Δ2
P3 = Δ1 + Δ2 + Δ3
And so on…
PT = Δ1 + Δ2 + Δ3 + … + ΔT


Pie Chart of Percent vs Type
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
Mushroom and Onion
9.2%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing

Percent

Frequency

Listing

Let PT = Δ1+ Δ2+…+ ΔT
be the total of the changes (variables) from
times (observations) 1 to T.
The sequence is
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
84/124
Summing

If the individual Δs are each normally
distributed with mean μ and standard
deviation σ, then
P1 = Δ1
= Normal [ μ, σ]
 P2 = Δ1 + Δ2
= Normal [2μ, σ√2]
 P3 = Δ1 + Δ2 + Δ3= Normal [3μ, σ√3]
 And so on… so that
 PT ~ N[Tμ, σ√T]

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
85/124
This Defines a Random Walk
The sequence is
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
90
200000
100000
15000
60
50
40
20000
22500
25000
IncomePC
27500
30000
32500
Histogram of Listing
30
6
200000
2
1
100000
15000
800000
1000000
369687
156865
51
80
8
5
400000
600000
Listing
Mean
StDev
N
10
500000
4
200000
Normal
100
12
700000
300000
0
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
400000
10
17500

600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
400000

900000
Mean
StDev
N
P-Value
95
500000

Scatterplot of Listing vs IncomePC
Normal - 95% CI
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball

P1 = Δ1
P2 = P1 + Δ2
P3 = P2 + Δ3
And so on…
PT = PT-1 + ΔT
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type

Percent

Meatball
Garlic 5.0%
2.3%
It follows that
Frequency

Listing

Percent

Mushroom and Onion
9.2%

P1 = Δ1
P2 = Δ1 + Δ2
P3 = Δ1 + Δ2 + Δ3
And so on…
PT = Δ1 + Δ2 + Δ3 + … + ΔT

Listing

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
86/124
A Model for Stock Prices
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

Preliminary:
Consider a sequence of T random
outcomes, independent from one to the
next, Δ1, Δ2,…, ΔT. (Δ is a standard symbol
for “change” which will be appropriate for
what we are doing here. And, we’ll use “t”
instead of “i” to signify something to do with
“time.”)
Δt comes from a normal distribution with
mean μ and standard deviation σ.
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
87/124
A Model for Stock Prices
Random Walk Model: Today’s price
= yesterday’s price + a change that is
independent of all previous
information.
 Start at some known P0 so
P1 = P0 + Δ1 and so on.
 Assume μ = 0 (no systematic drift in
the stock price).

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
88/124
Random Walk Simulations
Pt = Pt-1 + Δt, t = 1,2,…,100
Example: P0= 10, Δt Normal with μ=0, σ=0.02
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
89/124
Random Walk?
Dow Jones March 27 to May 26, 2011.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
90/124
Uncertainty
Expected Price = E[Pt] = P0+Tμ
We have used μ = 0 (no systematic
upward or downward drift).
 Standard deviation = σ√T reflects
uncertainty or “risk.”
 Looking forward from “now” = time
t=0, the uncertainty increases the
farther out we look to the future.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
91/124
Expected Range
[P0  t]  2 t
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2-C – Sums of
Random Variables
Part 2 – Expectations of Random Variables
93/124
Sequences of Independent
Random Variables
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

x1,x2,…,xn = a set of n random variables
 Same (marginal) probability distribution, f(x)
 Finite identical mean μ and variance σ2
 Statistically independent
IID = independent identically distributed
This is a “random sample” from the population f(x).
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
94/124
The Sample Mean
1 n
1
1
1
X

X

X

...

Xn

i
1
2
i 1
n
n
n
n
1 
1 
1

E[X]  E  X1   E  X 2   ...  E  X n 
n 
n 
n 
1
1
1
1
= E[X1 ]  ...  E[X n ]    ...  
n
n
n
n
= 
Sample Mean: X=
1
1
n
n
n
1 
Var[X]   i 1 Var  X i    i 1 2 Var  X i    i 1 2  2
n
n
n 
2
=
n
Distribution of X ? Remains to be seen.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
95/124
Convergence of a Random
Variable to a Constant
A constant, c, is a random variable, C, with variance zero.
The constant is always equal to c. Prob(C = c) = 1.
The sample mean, X has expected value .
2
The variance of X is
. X is not a constant when n is finite.
n
As n 
  X becomes a constant by this definition.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
96/124
Convergence in Mean Square
E[Xn ] = μ and
Var[Xn ]  0 as n  
Then Xn converges in
mean square to μ
If
If E[Xn ]  μ as n  
Var[Xn ]  0 as n  
Then Xn converges in mean
square to μ.
Xn + (1/n) converges in
mean square to μ.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
97/124
Histogram of Xbar_4
Normal
120
Mean
S tDev
N
100
10.03
1.763
1000
Frequency
80
60
40
20
0
4
6
8
10
Xbar_4
12
14
Histogram of Xbar_9
Normal
80
Mean
StDev
N
70
10.04
1.164
1000
Frequency
60
Convergence in Mean
Square: The top figure is a
histogram for 1,000 means
of samples of 4; the center is
for samples of 9, the lowest
one is for samples of 15.
The vertical bars go through
7, 10 and 13 on all three
figures.
50
40
30
20
10
0
6
7
Frequency
70
60
50
40
30
20
10
500000
400000
Mushroom
16.2%
Plain
32.5%
90
100000
15000
60
50
40
20000
22500
25000
IncomePC
27500
30000
32500
12.0
30
Histogram of Listing
6
200000
2
1
100000
15000
800000
1000000
Mean
StDev
N
369687
156865
51
80
8
5
400000
600000
Listing
Normal
10
500000
4
200000
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
100
12
700000
300000
0
12.8
14
800000
400000
10
17500
11.2
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
400000
9.6
10.4
Xbar_15
900000
Mean
StDev
N
P-Value
95
500000
8.8
Scatterplot of Listing vs IncomePC
Normal - 95% CI
600000
200000
8.0
Probability Plot of Listing
99
700000
300000
100000
7.2
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
600000
10.06
0.9054
1000
Percent
800000
800000
13
Mean
StDev
N
Frequency
Sausage
5.8%
900000
12
80
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
11
90
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
10
Xbar_9
Normal
Percent
Pie Chart of Percent vs Type
9
Histogram of Xbar_15
0
Mushroom and Onion
9.2%
8
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
98/124
Convergence of Means
If x1 ,...,x n is a random sample from a population with finite constant,
mean  and finite constant variance, 2 , then X converges in mean
square to 
Applies to functions of X. E.g., if Y = exp(X) and E[exp(X)] and
Var[exp(X)] are finite constants, then Y converges in mean square
to E[exp(X)]. If Xi ~ N[,2 ] and Y = exp(X) then Y 
 exp( + 12 2 ).
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
700000
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
Frequency
98
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
99/124
Probability Limits: Plim xn
Let  be a constant,  be any positive value,
and n index the sequence.
If lim(n  )Prob[|x n   | > ]  0 then,
x n converges in probability to .
In words, the probability that the difference
between x n and  is larger than  for any 
goes to zero. x n becomes arbitrarily close to .
If lim(n  )Prob[|x n   | > ]  0 then plim x n  .
Mean square convergence is sufficient (not necessary)
for convergence in probability.
Mean square convergence is sufficient for most work
in applied statistics.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
100/124
Probability Limits and
Expectations
What is the difference between
E[xn] and plim xn?
Consider: X = n with prob(X=n)=1/n
X = 1 with prob(X=1)=1 – 1/n
E[X]=2 – 1/n  2
Plim(X)
=1
A notation
P
plim xn    x n 

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
101/124
The Slutsky Theorem
Assumptions: If
xn is a random variable such that plim xn = θ.
For now, we assume θ is a constant.
g(.) is a continuous function with continuous
derivatives. g(.) is not a function of n.
Conclusion: Then plim[g(xn)] = g[plim(xn)]
assuming g[plim(xn)] exists.
Works for probability limits. Does not work for
expectations.
E[xn ]=; plim(xn )  , E[1/xn ]=?; plim(1/xn )=1/
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
102/124
Multivariate Slutsky Theorem
Plim xn = a, Plim yn = b
 g(xn,yn) is continuous, has continuous
first derivatives and exists at (a,b).
 Plim g(xn,yn) = g(a,b)
 Generalizes to K functions of M
random variables

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
103/124
Monte Carlo Integration
1 n
p
Since  i 1 g ( xi ) 
E[ g ( x)], a random sample can be used
n
to approximate the expected value. Two cases:
(1) The population is known: Randomly draw R observations from
1 R
p
the known population, x1 ,...,x R .
g
(
x
)

E[ g ( x)]. This is

r
r 1
R
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
g ( x) f ( x) dx.
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball

Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

Frequency
equivalent to estimating

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
104/124
Monte Carlo Integration
The population is unknown or the limits of integration are finite.
Compute

b
a
g(x)dx
Strategy: Let z = (x-a)/(b-a) so x = a + (b-a)z and
dx = (b-a)dz and z ranges from 0 to 1

b
a
1
1
0
0
g(x)dx   g[a+(b-a)z](b-a)dz (b-a)  g[a+(b-a)z]dz
Draw a sample from z ~ U[0,1].
Average R draws on g[a + (b-a)z] then multiply by (b-a).
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
105/124
Application
Normal probability from
-1 to +1.5 is
0.3413 + 0.4332 = .7745.
[a = -1, b = +1.5, g(z)=(z).]
Compute 10,000 random draws on x from U[0,1].
Compute z = a + (b-a)x = -1+2.5*x
Average the 10,000 draws on (z) then multiply by
the average by (b-a) = (1.5 – (-1)) = 2.5.
Gives .773641 in my experiment.
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

Percent

Listing

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
106/124
Application
For Normal(2,1.52),
E[exp(x)] = exp(2 + ½1.52) = 22.76
 Draw 10,000 random U(0,1) draws.
Transform to x ~ N(0,1) then
z = 2 + 1.5*x
 Compute q=exp(z) and average 10,000
draws on q.
 My result was 22.87944.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
107/124
Limit Results
Mean converges in probability to . Variance
goes to zero.
If n is finite, what can be said about its
behavior?
Objective: characterize the distribution of the
mean when n is large but finite
Strategy: find a limit result then use it to
approximate for finite n.
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Percent

Frequency

Listing

Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
108/124
A Finite Sample Distribution
180
F req u en cy
135
90
45
0
. 208
. 288
. 367
. 447
. 526
. 606
. 686
. 765
XB8
Means of 1000 samples of 8 observations from U[0,1].
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
109/124
Central Limit Theorems
Set of independent random variables, X1 ,..., X n . Same distribution, f(X),
not necessarily normal.
2
X n 
 , Var[X n ]= 
 0. Large sample behavior is obvious.
n
X
Stabilize X n . Let Zn =
.
/ n
Has E[Zn ]=0 and Var[Zn ]  1 for every n, even if n is huge (infinite).
What can be said about the probability distribution of Zn ?
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
110/124
Limiting Distributions
Xn has probability density fXn(Xn) and
cdf FXn(Xn).
 If FXn(Xn)  F(X) as n  , then F(X)
is the limiting distribution. (At points
where F(X) is continuous in X.)

FXn (X) 
 F(X) implies that Xn converges in distribution to X.
d
Written Xn 
X
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
111/124
Lindeberg – Levy Central
Limit Theorem
Set of independent random variables, X1 ,..., X n . Same distribution, f(X),
not necessarily normal. X n 
1 n
X

i 1 i
n
2
X n 
 , Var[X n ]= 
 0. Large sample behavior is obvious.
n
X
d
Stabilize X n . Let Zn =
. Z n 
 N [0,1].
/ n
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
112/124
Other Central Limit Theorems
Lindeberg Levy for i.i.d.
 Lindeberg Feller – heteroscedastic.
Variances may differ
 Lyupanov: distributions may differ
 Extensions - time series with some
covariance across observations.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
113/124
Rough Approximations
d
Z n 
 N [0,1].
Requires infinite n. Suppose n is fairly large. Assume this applies
approximately, and manipulate it.
Zn 


X
/ n

 N [0,1]. Assume it holds approximately for finite n.
 2 
 / n Zn  X   is approximately N  0, 
 n 
 2 
 / n Zn    X is approximately N  , 
 n 




n
n   / n Zn      i 1 X i is approximately N  n, n 2 


600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
114/124
Normal Approximation to Binomial
Binomial (n,p) equals the sum of n
Bernoulli’s with parameter p.
 Each Bernoulli X has  = p and
2 = p(1-p).
 Sum of n variables is approximately
normal with mean np and variance
np(1-p).

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
115/124
Approximation to binomial
with n = 48, p=.25
Prob[8  x  15]= P(X=8)+P(X=9)+...+P(X=15)
 48  x
=  X=8  .25 (1  .25)48 x
 x 
= 0.815678
15
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
116/124
Demoivre’s Normal Approximation
The binomial density
function has n=48,
θ=.25, so
μ = 12 and σ = 3.
The normal density
plotted has mean 12
and standard
deviation 3.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
117/124
Using deMoivre’s Approximation
8 0.057905
9 0.085785
10 0.111520
11 0.128417
12 0.131984
13 0.121832
14 0.101526
15 0.076709
Total
0.815678
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Marginal Plot of Listing vs IncomePC
Normal
100
12
700000
400000
30

Empirical CDF of Listing
14
800000
600000
70
20
300000
100000
15000
369687
156865
51
0.994
0.012
80
400000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
500000
200000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
600000
200000
0.7495859
8.1% error
What
happened?
Probability Plot of Listing
99
700000
300000
0.8413450 – 0.0917591 =
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
P[z < 1] – P[z < -1.33]=
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
P[-1.33 < z < 1]=
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
P[(8-12)/3<z<(15-12)/3]=
Frequency
The binomial has
n=48, θ=.25, so
μ = 12 and σ = 3. The
normal distribution
plotted has mean 12
and standard
deviation 3.
P[8 < x < 15]=
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
118/124
Continuity Correction
When using a continuous distribution
(normal) to approximate a discrete
probability (binomial), subtract .5 from
the lowest value in the range and add
.5 to the highest value in the range.
(The correction becomes less
important as n increases.)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
119/124
Correcting deMoivre’s Approximation
8 0.057905
9 0.085785
10 0.111520
11 0.128417
12 0.131984
13 0.121832
14 0.101526
15 0.076709
Total
0.815678
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
95
600000
200000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
Mean
StDev
N
P-Value

0.8115198
0.5% error
Probability Plot of Listing
99
700000
300000
0.878327 – 0.0668072 =
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
P[z < 1.166] – P[z < -1.5]=
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
P[-1.5 < z < 1.166]=
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
P[(7.5-12)/3<z<(15.5-12)/3]=
Frequency
The binomial has
n=48, θ=.25, so
μ = 12 and σ = 3. The
normal distribution
plotted has mean 12
and standard
deviation 3.
P[7.5 < x < 15.5]=
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
120/124
A Useful Convergence Result
If X n converges to X and if g(X n ) is a continuous function,
then Yn = g(X n ) converges to Y = g(X).
d
Example: If X n 
 N[0,1] then X 2n converges to the random
variable that is the square of a standard normal, which is Gamma  12 , 12  .
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
121/124
Combine Slutsky with the
Central Limit Theorem

Zn 
X
/ n
General Result: if Xn()  F(X|)
and if yn  , then Xn(yn)  F(X|)

 N [0,1]
Suppose s n 
1
n 1
 i 1  Xi -X 
n
2
p

.
Then,
X
Zˆ n 

 N [0,1].
sn / n
Replacing  with a random variable that converges to  preserves the limiting result.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
122/124
Asymptotic Distributions
An asymptotic distribution is an
approximation to a true finite n
distribution based on a result found for
the limiting distribution (with infinite n)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
123/124
Asymptotic Distribution
Infinite n result
Zn 
X

 N [0,1] as n 

/ n
Assume it is approximately true for finite n.
Zn 
X
~ N [0,1] for finite n
/ n
Manipulate as before
 2 
X ~ N  ,
 approximately.
 n 
This is the "asymptotic" distribution of X. This is how we use the
central limit to understand the sampling distribution of X.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 2 – Expectations of Random Variables
124/124
The Chebychev Inequality
For any random variable with finite mean
 and variance 2,
 Prob[|X-|/ > k] < 1/k2
 Prob X is farther than k standard
deviations from the mean is less than or
equal to 1/k2.
 Useful for proofs, not for practical
computations.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
0
Mean
StDev
N
10
500000
300000
10
Normal
100
12
700000
400000
30
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
C ategory
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
```