Download Gamma & Beta Distributions - Lyle School of Engineering

Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340 :
PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Special Continuous Probability
Distributions
Gamma Distribution
Beta Distribution
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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Gamma Distribution
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The Gamma Distribution
• A family of probability density functions that yields
a wide variety of skewed distributional shapes is the
Gamma Family.
• To define the family of gamma distributions, we first
need to introduce a function that plays an important
role in many branches of mathematics, i.e., the Gamma
Function
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Gamma Function
•
Definition
For
  0 , the gamma function ( )is defined by

( )   x 1e  x dx
0
•
Properties of the gamma function:
1. For any   1, ( )  (
[via integration by parts]
2. For any positive integer,
3.
1
  
2
 1)  (  1)
n, (n)  (n  1)!

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Family of Gamma Distributions
• The gamma distribution defines a family of which
other distributions are special cases.
• Important applications in waiting time and reliability
analysis.
• Special cases of the Gamma Distribution
– Exponential Distribution when α = 1
– Chi-squared Distribution when


2
and   2,
Where  is a positive integer
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Gamma Distribution - Definition
A continuous random variable Xis said to have a gamma distribution
if the probability density function of X is
f ( x;  ,  ) 
1
 1
x e

 (  )

x

for
x  0,
otherwise,
0
where the parameters and  satisfy
  0,   0.
The standard gamma distribution has
 1
The parameter  is called the scale parameter because values other
than 1 either stretch or compress the probability density function.
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Standard Gamma Distribution
The standard gamma distribution has   1
The probability density function of the standard
Gamma distribution is:
1  1  x
f ( x; ) 
x e
( )
for x  0
And is 0 otherwise
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Gamma density functions
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Standard gamma density functions
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Probability Distribution Function
If
X~ G ( ,  ), then
the probability distribution function of X is

1
 1  y
*
F ( x)  P( X  x) 
y
e
dy

F
( y;  )

( ) 0
for y=x/β and x ≥ 0.
Then use table of incomplete gamma function in
Appendix A.24 in textbook for quick computation of
probability of gamma distribution.
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Gamma Distribution - Properties
If x ~ G ( ,  ) , then
•Mean or Expected Value
  E (X )  
•Standard Deviation
  
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Gamma Distribution - Example
Suppose the reaction time X of a randomly selected
individual to a certain stimulus has a standard
gamma distribution with α = 2 sec. Find the
probability that reaction time will be
(a) between 3 and 5 seconds
(b) greater than 4 seconds
Solution
Since
P(3  X  5)  F (5)  F (3)  F * (5; 2)  F * (3; 2)
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Gamma Distribution – Example (continued)
Where
and
3
1
F (3;2)  
ye  y dy  0.801
2 
0
*
5
1
y
F (5;2)  
ye dy  0.960
2 
0
*
P(3  x  5)  0.960  0.801  0.159
The probability that the reaction time is more than
4 sec is
P( X  4)  1  P( X  4)  1  F * (4; 2)  1  0.908
 0.092
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Incomplete Gamma Function
Let X have a gamma distribution with parameters
Then for any x>0, the cdf of X is given by
 and  .
x
P( X  x)  F ( x; ,  )  F ( ; )
*

x
Where F ( ;  ) is the incomplete gamma function.

*
MINTAB and other statistical packages will calculate F ( x;  ,  )
once values of x,  , and  have been specified.
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Example
Suppose the survival time X in weeks of a randomly selected male
mouse exposed to 240 rads of gamma radiation has a gamma
distribution with   8 and   15
The expected survival time is E(X)=(8)(15) = 120 weeks
and
  (8)(152 )  42.43 weeks
The probability that a mouse survives between 60 and 120 weeks is
P(60  X  120)  P( X  120)  P( X  60)
 F (120;8,15)  F (60;8,15)
 F * (8;8)  F * (4;8)
 0.547  0.051
 0.496
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Example - continue
The probability that a mouse survives at least 30 weeks is
P( X  30)  1  P( X  30)  1  P( X  30)
 1  F (30;8,15)
 1  F (2;8)
 1  0.001
*
 0.999
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Beta Distribution
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Beta Distribution - Definition
A random variable X is said to have a beta distribution
with parameters, ,
 , A , and B
if
the probability density function of X is
f ( x ;  ,  , A, B )
 1
 1
(  +  )  x  A   B  x 
1



 

B  A (  )  (  )  B  A   B  A 
for A  x  B
,
and is 0 otherwise,
where
  0,   0
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Standard Beta Distribution
If X ~ B(  ,  , A, B), A =0 and B=1, then X is said to have a
standard beta distribution with probability density function
( +  )  1
f ( x; ,  ) 
x (1  x)  1
( )(  )
for
0  x 1
and 0 otherwise
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Graphs of standard beta probability density function
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Beta Distribution – Properties
If X ~ B(  ,  , A, B),
then
•Mean or expected value
  A +  B  A 

 +
•Standard deviation
B  A 
 
 +    +  + 1
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Beta Distribution – Example
Project managers often use a method labeled PERT for
Program Evaluation and Review Technique to coordinate
the various activities making up a large project. A
standard assumption in PERT analysis is that the time
necessary to complete any particular activity once it has
been started has a beta distribution with A = the
optimistic time (if everything goes well) and B = the
pessimistic time (If everything goes badly). Suppose that
in constructing a single-family house, the time X (in
days) necessary for laying the foundation has a beta
distribution with A = 2, B = 5, α = 2, and β = 3. Then
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Beta Distribution – Example (continue)

 .4 , so E ( X )  2 + (3)(0. 4)  3.2 . For these values of α
 +
and β, the probability density functions of X is a simple
polynomial function. The probability that it takes at most
3 days to lay the foundation is
1 4!  x  2  5  x 
P( X  3)    

 dx
3 1!2!  3  3 
2
2
3
3
4
4 11 11
2
  x  25  x      0.407 .
27 2
27 4 27
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