### Binomial, Zeng 1.4

```MAT 4830
Mathematical Modeling
Section 1.4
Conditional Statements
http://myhome.spu.edu/lauw
Preview
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Review Binomial Distribution
Introduce the first type of repetition
statements – the for loop
Allow a specific section of code to be
executed a number of times
Introduces simple arrays
Example 0
Suppose that there are  devices, each
with a probability  of failure during a given
time period. What is the probability that
exactly  fail during this time period?
n devices
F
P( failure)  p
Example 0
Suppose that there are  devices, each
with a probability  of failure during a given
time period. What is the probability that
exactly  fail during this time period?
n devices
F
F
k devices fail
F
P( failure)  p
Example 0
r.v. =no. of devices fail
P( X  k )  ?
n devices
F
F
k devices fail
F
P( failure)  p
P( X  k )  ?
Example 0
n devices
n  k devices
k devices
F
F
F
P( X  k )  ?
Example 0
n devices
n  k devices
k devices
F
F
p
k
F
(1  p)
n k
n
 
k 
Example 0
n devices
n  k devices
k devices
F
F
F
n k
nk
P( X  k )    p (1  p)
k 
Binomial Distribution B(n,p)
X
B ( n, p )
n k
Prob. Mass Fun. f (k )  P( X  k )    p (1  p) n  k , k  0,1,..., n
k 
n x
Alternatively,
f ( x)  P( X  x)    p (1  p) n  x , x  0,1,..., n
 x
Mean   EX  np
Std. D.   np(1  p)
Binomial Distribution B(n,p)
X
B ( n, p )
n k
Prob. Mass Fun. f (k )  P( X  k )    p (1  p) n  k , k  0,1,..., n
k 
n x
Alternatively,
f ( x)  P( X  x)    p (1  p) n  x , x  0,1,..., n
 x
Mean   EX  np
Std. D.   np(1  p)
Binomial Distribution B(n,p)
X
B ( n, p )
n k
Prob. Mass Fun. f (k )  P( X  k )    p (1  p) n  k , k  0,1,..., n
k 
n x
Alternatively,
f ( x)  P( X  x)    p (1  p) n  x , x  0,1,..., n
 x
Mean   EX  np
Std. D.   np(1  p)
Team HW #1
Team Homework #1
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Use the definition of expected value and the
binomial theorem
 n  k nk
( a  b)     a b
k 0  k 
n
n
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Do not use the moment generating function.
You may need to recall how to shift indices in
a summation (see the hidden slides below for
review).
Team Homework #2
A campaign staff knows from experience
that only one in every three volunteers
called will actually show up to distribute
leaflets.
Team Homework #2
How many phone calls must be made to
guarantee at least 20 workers with a
confidence of 90%?
Team Homework #2
Minimum
How many phone calls must be made to
guarantee at least 20 workers with a
confidence of 90%?
P(at least 20 workers)  0.9
Team Homework #2
Use a binomial model to solve the problem.
 You need to write a Maple program to help
you solve the problem.
 You need to explain your methodologies,
arguments, and conclusions carefully.
 Extra works are welcome – In the past,
students had done more than they were
Zeng Section 1.4
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Introduce the first type of repetition
statements – the for loop
Allow a specific section of code to be
executed/repeated a number of times
Introduces simple arrays
Zeng Section 1.4
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Please listen to the explanations before
you type in the program.
It takes one minute to explain.
Example 1
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Print the square of the first 10 positive
integers
What is the task being repeated?
Example 1
>sq:=proc()
local i;
for i from 1 to 10 do
print(i^2);
od;
end:
#program to print the square
#of the 1st 10 positive
#integers
#index
#A loop to print the integers
#output i^2
Example 1
>sq:=proc()
#program to print the square
#of the 1st 10 positive
#integers
#index
local i;
for i from 1 to 10 do
print(i^2);
od;
end:
i
i
i
2
#A loop to print the integers
#output i^2
1
1
2
4
10
100
Example 1
>sq:=proc()
local i;
for i from 1 to 10 do
print(i^2);
od;
end:
#program to print the square
#of the 1st 10 positive
#integers
#index
#A loop to print the integers
#output i^2
> sq();
1
4
9
Structure of the for loop
for loop_index from start_value to end_value do
block of statements to be repeated
od;
Structure of the for loop
for loop_index from start_value to end_value do
block of statements to be repeated
od;
The loop_index increase by the default step
size 1 everytime the execution of block of
statements to be repeated is finished.
Different step size can be used by adding “by
stepsize” feature.
Example 2
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Print the square of the first 10 positive
odd integers
Example 2
Example 2
> sq2();
1
9
25
Example 3
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Print the square of the first  positive
integers
Example 3
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Print the square of the first  positive
integers
Introduces array and seq
Note that these commands are not
necessary here
Example 3
Example 3
x[1]
x[2]
x[3]
x[n]
Example 3
> sq3(2);
1, 4
> sq3(5);
1, 4, 9, 16, 25
Example 4
Fibonacci sequence is defined by
F0  0, F1  1, Fk  Fk 1  Fk 2 for k  2,3,
{0, 1, 1, 2, 3, 5,
}
Example 4
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F0  0, F1  1, Fk  Fk 1  Fk 2
Write a program that generate the first
+ 1 terms of the Fibonacci sequence
0, 1, … ,
Example 4
Why there is no
print statement?
F0  0, F1  1, Fk  Fk 1  Fk 2
Example 4
F0  0, F1  1, Fk  Fk 1  Fk 2
Example 5
(1)
2 k 1
sin x  
x

k  0 (2k  1)!

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k
(1)
2 k 1
x

k  0 (2k  1)!
n
k
Write a program, for the input of  and ,
to approximate the value of sin() by the
first sum of the first  + 1 terms in the
Taylor series.
Example 5
(1)
2 k 1
sin x  
x

k  0 (2k  1)!

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k
(1)
2 k 1
x

k  0 (2k  1)!
n
k
This is to demonstrate the basic form of
“accumulation”.
(1) k 2 k 1
sin x  
x
k  0 (2 k  1)!
n
Example 5
(1) k 2 k 1
sin x  
x
k  0 (2 k  1)!
n
Example 5
(1) k 2 k 1
sin x  
x
k  0 (2 k  1)!
n
Example 5
Homework
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See course webpage