Linear Combinations

Warm Up
• How do I know this is a probability
distribution?
• What is the probability that Mary hits
x=# red
lights
p(x)
0
0.05
1
0.25
2
0.35
3
0.15
4
0.15
5
0.05
exactly 3 red lights?
• What is the probability that she gets
at least 4 red lights?
• What is the probability that she gets
less than two?
• Find the mean & standard deviation.
Find Mean & Standard Deviation:
x=#
books
P(x)
0
0.13
1
0.21
2
0.28
3
0.31
4
0.07
Ex.
1.
2.
3.
x = possible
winnings
P(x)
5
0.1
7
0.31
8
0.24
10
0.16
14
0.19
Find the mean
Find the Standard Deviation
Find the probability that x is within one
deviation from the mean.
500 raffle tickets are sold at \$2 each. You bought 5 tickets.
What’s your expected winning if the prize is a \$200 tv.?
There are four envelopes in a box. One envelope contains a \$1
bill, one contains a \$5, one contains a \$10, and one a \$50 bill. A
person selects an envelope. Find the expected value of the
draw. What should we charge for the game for it to be fair?
A person selects a card from a deck. If it is a red card, he wins
\$1. If it is a black card between or including 2 and 10, he wins
\$5. If it is a black face card, he wins \$10, and if it is a black ace,
he wins \$100. Find the expectation of the game. What would it
be if it cost \$10 to play? What should I charge to make it a fair
game?
On a roulette wheel, there are 38 slots numbered 1 through 36
plus 0 and 00. Half of the slots from 1 to 36 are red; the other
half are black. Both the 0 and 00 slots are green. Suppose that
a player places a simple \$1 bet on red. If the ball lands in a red
slot, the player gets the original dollar back, plus an additional
dollar for winning the bet. If the ball lands in a different-colored
slot, the player loses the dollar bet to the casino. What is the
player’s average gain?
LINEAR
TRANSFORMATIONS
Section 6.2A
Remember – effects of Linear
Transformations
• Adding or Subtracting a Constant
• Adds “a” to measures of center and location
• Does not change shape or measures of spread
• Multiplying or Dividing by a Constant
• Multiplies or divides measures of center and location by “b”
• Multiplies or divides measures of spread by |b|
• Does not change shape of distribution
Adding/Subtracting a constant from data shifts the mean but
doesn’t change the variance or standard deviation.
•
E  X  c  E(X )  c
•
Var  X  c   Var ( X )
Multiplying/Dividing by a constant multiplies the mean and
the standard deviation.
E ( aX )  aE ( X )
aX = a ∙
Var ( aX )  a Var ( X )
2
Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The
vehicle will hold up to 6 passengers. The number of passengers X
on a randomly selected day has the following probability
distribution. He charges \$150 per passenger. How much on average
does Pete earn from the half-day trip?
# Passengers
Prob
2
0.15
3
0.25
4
0.35
5
0.2
6
0.05
Pete’s Jeep Tours offers a popular half-day trip in a tourist area. The
vehicle will hold up to 6 passengers. The number of passengers X
on a randomly selected day has the following probability
distribution. He charges \$150 per passenger. What is the typical
deviation in the amount that Pete makes?
# Passengers
Prob
2
0.15
3
0.25
4
0.35
5
0.2
6
0.05
What if it costs Pete \$100 to buy permits, gas, and a ferry pass for
each half-day trip. The amount of profit V that Pete makes from the
trip is the total amount of money C that he collects from the
passengers minus \$100. That is V = C – 100. So, what is the average
profit that Pete makes? What is the standard deviation in profits?
A large auto dealership keeps track of sales made during each hour
of the day. Let X = the number of cars sold during the first hour of
business on a randomly selected Friday. Based on previous records,
the probability distribution of X is shown below. Suppose the
dealership’s manager receives a \$500 bonus from the company for
each car sold. What is the mean and standard deviation of the
amount that the manager earns on average?
# cars sold
Prob
0
0.3
1
0.4
2
0.2
3
0.1
Suppose the dealership’s manager receives a \$500 bonus
from the company for each car sold. To encourage customers
to buy cars on Friday mornings, the manager spends \$75 to
provide coffee and doughnuts. Find the mean and standard
deviation of the profit the manager makes.
# cars sold
Prob
0
0.3
1
0.4
2
0.2
3
0.1
Variance of y = a + bx
• Relates to slope.
y
b
x
b 
2
y
2
x
2
y  b  x
2
2
var( y )  b

2
y
b 
2
2
x
2
2
 var( x ) 
Effects of Linear Transformation on the
Mean and Standard Deviation if  =
+ .
=  +
=
*Shape remains the same.
Example: Three different roads feed into a freeway entrance. The number of
cars coming from each road onto the freeway is a random variable with mean
values as follows. What’s the mean number of cars entering the freeway.
Mean #
Cars
1
800
2
1000
3
600
Mean of the Sum of Random Variables
For any two random variables, X and Y, if  =  +
then the expected value of T is
=   =  +
Ex: What is the standard deviation of the # of cars coming
from each road onto the freeway.
Mean #
Cars
St.
Dev.
1
800
34.5
2
1000
42.8
3
600
19.3
Variance of the Sum of Random
Variables
For any two random variables, X and Y, if  =  +
then the variance of T is
x
y
 x y 
3x2 y 
Mean
20
24
st dev
5
3
x
y
 x y 
3x y 
Mean
20
24
st dev
5
3
x
y
 x y 
 3x2 y 
Mean
20
24
st dev
5
3
x
y
 x y 
 3x y 
Mean
20
24
st dev
5
3
Find: 
and 
3x2 y
3x2 y
x
P(x)
y
P(y)
3
0.32
10
0.22
4
0.14
20
0.34
5
0.12
30
0.18
6
0.42
40
0.26
Homework
Worksheet