# Gamma Beta

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```					                                    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
1
Gamma Distribution

2
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

3
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, ( )  (          1)  (  1)
[via integration by parts]

2. For any positive integer,     n, (n)  (n  1)!
1
3.            
2
4
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

         and   2,
2
Where  is a positive integer

5
Gamma Distribution - Definition

A continuous random variable Xis said to have a gamma distribution
if the probability density function of X is

x

1       1        
x e              for   x  0,
f ( x; ,  )       (  )


0                       otherwise,

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.
6
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

7
Gamma density functions

8
Standard gamma density functions

9
Probability Distribution Function

If   X~ G ( ,  ), then
the probability distribution function of X is

1
( ) 
F ( x)  P( X  x)          y 1e  y 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.

10
11
Gamma Distribution - Properties

If x ~ G ( ,  ) , then

•Mean or Expected Value

  E (X )  

•Standard Deviation

  

12
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)

13
Gamma Distribution – Example (continued)
3
Where                 1
F (3;2)  
*
ye  y dy  0.801
0
2 
and                 5
1
F (5;2)  
*                 y
ye dy  0.960
0
2 

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
14
Incomplete Gamma Function
Let X have a gamma distribution with parameters     and  .
Then for any x>0, the cdf of X is given by
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.
15
Example

Suppose the survival time X in weeks of a randomly selected male
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
16
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

17
Beta Distribution

18
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
1    (  +  )  x  A   B  x 
                                                   ,
B  A (  )  (  )  B  A   B  A 
for A  x  B
and is 0 otherwise,
where       0,   0
19
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         1
f ( x;  ,  )              x (1  x)           for   0  x 1
( )(  )

and 0 otherwise

20
Graphs of standard beta probability density function

21
Beta Distribution – Properties

If X ~ B(  ,  , A, B),   then

•Mean or expected value


  A +  B  A 
 +
•Standard deviation

B  A 
 
 +    +  + 1
22
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

23
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 
3                           2

P( X  3)                      dx
2
3 1!2!  3  3 
3
  x  25  x      0.407 .
4                2  4 11 11
27 2                27 4 27
24

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