Download EVERYDAY ENGINEERING EXAMPLES FOR SIMPLE CONCEPTS

EVERYDAY
ENGINEERING
EXAMPLES FOR SIMPLE
CONCEPTS
ENGR 3200 – Probability and Statistics
Dr. Heriberto Barriera
Copyright © 2015
Discrete
Probability
Distribution
– Negative
Binomial
MSEIP – Engineering
Everyday Engineering Examples
An Easy Way to Demonstrate Negative Binomial Distribution
Engage:
In a small group, roll a die until a “1” is observed. Repeat this process a total of
30 times and record the number of rolls required to get a “1” in each of the 30
trials of the experiment.
Trial #
1
2
3
4
5
6
7
8
9
10
Number of
rolls to get
a “1”
Trial #
Number
of rolls to
get a “1”
11
12
13
14
15
16
17
18
19
20
Trial #
Number
of rolls to
get a “1”
21
22
23
24
25
26
27
28
29
30
Use your results to obtain a point estimate of the mean and compare the results
with the expected value of the number of rolls of a die required to obtain a “1”
using the formula for the negative binomial distribution.
Explore:
The sample space of an experiment, denoted S, is the set of all possible outcomes
Sample Space: S ={1, 2, 3, 4, 5, 6}
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Outcomes: landing with a 1, 2, 3, 4, 5, or 6 face up.
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of that experiment. The number of possible outcomes when rolling a die are:
You can begin rolling a die until a “1” is observed. As you can see, the number of
times necessary until a “1” is observed, is different. The probability to get a “1” in
a fair die is 1/6. As the number of time rolling a die is increased, the average of
times to get a ”1” is approximated.
Explain:
A negative binomial experiment is a statistical experiment that has the following
properties:

The experiment consists of x repeated trials.

Each trial can result in just two possible outcomes. We call one of these
outcomes a success and the other, a failure.

The probability of success, denoted by P, is the same on every trial.

The trials are independent; that is, the outcome on one trial does not
affect the outcome on other trials.

The experiment continues until r successes are observed, where r is
specified in advance.
Elaborate:
This experiment is a negative binomial experiment because:

The experiment consists of repeated trials. We roll a die repeatedly until it a
“1” is observed.

Each trial can result in just two possible outcomes – success “1” or failure
“2,3,4,5,6”.

The probability of success is constant – 1/6 on every trial.

The trials are independent; that is, getting a “1”on one trial does not affect
The experiment continues until a fixed number of successes have occurred;
in this case, a “1”.
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
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whether we get a “1” on other trials.
The pmf of the negative binomial rv X with parameters r = number of S’s and
p = P(S) is
 x  r  1 r
 p 1  p x
nb( x; rp)  
 r 1 
Where x = 0, 1, 2, … and represent the number of failures necessary to get the
r success
The expected value and the variance for the negative binomial distribution is
E( X ) 
r (1  p)
p
V (X ) 
r (1  p)
p2
Evaluate:
Invite students to attempt the following problem:
Example:
A basketball player made 45% of his shots from the three point line. Find the probability
that this player will get his 5 made on the 9 attempt.
Using the negative binomial distribution:
x = 4, that represents the number of failures
p = 0.45, that represents the probability of success
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 x  r  1 r
 p 1  p x
nb( x; rp)  
 r 1 
3
r = 5, that represents the number of success
 4  5  1
8 
(.45)5 (1  .45) 4   (.45)5 (1  .45) 4  0.1182
nb(4;5,.45)  
 5 1 
 4
What is the expected number of attempts to get the 5 made shots?
E ( x) 
r (1  p)
p
E ( x) 
5(1  .45)
6
.45
Therefore the number of attempts necessary to get 5 shots made are 11 (5 (the number
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4
of success) + 6 (the number of failures)).