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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
                                                          Leadership in Engineering
                                                                              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
 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
                                                             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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