Discrete Distribution Functions
Jake Blanchard
Spring 2010
Uncertainty Analysis for Engineers
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Binomial Distributions
Many problems involve repeated,
independent trials in which each trial has
two possible outcomes and probability of
success is a constant
 Best example is coin tossing

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Other examples
Ten space shots are planned. Probability
of success is 0.95. What are chances that
at least 9 are successful?
 10 numbers are rounded to nearest
integer. What is probability that half are
rounded up and half down?
 20 sales presentations are planned. The
probability of receiving orders is 1/3.
What is the probability of getting at least
10 orders?

Uncertainty Analysis for Engineers
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Binomial distribution
The probability of x successes and (n-x)
failures in n trials is px(1-p)n-x, where p is
the probability of success on a single trial
 The probability of exactly x successes in n
independent trials is

n x
n x
f ( x)    p 1  p 
 x
n
n!
  
 x  x!n  x !
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Binomial Distribution

The binomial cdf is thus
n x
n x
F (r )     p 1  p 
x 0  x 
r
E(x)=np
 Var(x)=np(1-p)

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Plots of pdf for n=10
0.3
0.25
p=0.25
0.2
0.15
0.3
0.1
0.25
0.05
0.2
0
0.15
0
1
2
3
4
5
6
7
8
9
10
p=0.5
0.1
0.05
0.3
0
1
0.25
2
3
4
5
6
7
8
9
10
11
p=0.75
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
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Example

Select 20 units at random from a
manufacturing lot. The lot is accepted if
there are less than 4 defective units
among the 20. If the average defect rate is
10%, what is the probability of
acceptance?
3
F (3)   0.1x 0.9 20 x  0.867
x 0

binocdf(3,20,0.1)
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Hypergeometric Distribution
This is the probability of exactly x
successes in n trials when n is not small
relative to the population size
 Useful when sampling from a small lot
 For N=total units, k=defective units in
sample, n=sample size, x=successes

 k  N  k 
 

x  n  x 

f ( x) 
N
 
n
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Geometric Distribution

The probability that (x-1) failures will be
followed by a success is
f ( x)  (1  p)

x 1
p
The probability that the first failure will
occur on the xth trial is
f ( x)  p
x 1
(1  p)
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The Poisson Distribution
Question: What is the probability that
exactly x customers will enter a store
during a particular time interval?
 Binomial distribution cannot help us, in
part because we don’t know what to use
for a sample size. Also, some customers
might come more than once.
 So instead of using a probability per trial,
we can use data for the average number
of arrivals.

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Poisson distribution
Distribution describes number of events
occurring within a fixed time interval
 For example, number of counts on a
radiation detector
 If, in a time interval, we expect  events,
then P is the probability that there are
exactly k occurrences within an interval
 k exp   
P(k ) 
k  0,1, 2, ...
k!

Uncertainty Analysis for Engineers
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Poisson Distribution
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Uses
Number of alpha particles emitted by a
source in a specified time interval
 Number of incoming calls per minute on
a switchboard
 Number of insurance claims per year
 Number of flaws in similar pieces of
material
 Number of bacteria on a slide
 Number of bombs falling on equal areas
of land in London

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Example
Flaws in a weld arise independently at a
rate of two flaws per weld
 What is the probability the weld has 0, 1,
2, or 3 defects?
 Solution: =2


x
0
1
2
3
#
.135
.271
.271
.18
eg, poisspdf(0,2)
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