1. Special Continuous Probability Distributions Leadership in 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

2. Gamma Distribution

3. 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

4. Gamma Function Definition For , the gamma function is defined by
Properties of the gamma function:
For any
[via integration by parts]
2. For any positive integer, 3.

5. 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
Where is a positive integer

6. Gamma Distribution - Definition
A continuous random variable is said to have a gamma distribution
if the probability density function of is
where the parameters and satisfy
The standard gamma distribution has
The parameter is called the scale parameter because values other
than 1 either stretch or compress the probability density function. , ) ( 1 G - x
for e b a
otherwise,

7. Standard Gamma Distribution
The standard gamma distribution has
The probability density function of the standard Gamma distribution is:
for
And is 0 otherwise

8. Gamma density functions

9. Standard gamma density functions

10. Probability Distribution FunctionIf
the probability distribution function of is
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. ~

12. Gamma Distribution - Properties
If x ~ G
, then
Mean or Expected Value
Standard Deviation

13. Gamma Distribution - Example
Suppose the reaction time 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

14. Gamma Distribution – Example (continued)
Where
and
The probability that the reaction time is more than
4 sec is

15. Incomplete Gamma Function
Let X have a gamma distribution with parameters and .
Then for any x>0, the cdf of X is given by
Where is the incomplete gamma function.
MINTAB and other statistical packages will calculate once values of x, , and have been specified.

16. 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 and
The expected survival time is E(X)=(8)(15) = 120 weeks
and
weeks
The probability that a mouse survives between 60 and 120 weeks is

17. Example - continue
The probability that a mouse survives at least 30 weeks is

18. Beta Distribution

19. Beta Distribution - Definition
A random variable is said to have a beta distribution
with parameters, , , and if
the probability density function of is , ) ( 1 ;
otherwise, is
and B x A
for f ÷ ø ö ç è æ - G × + = b a
where

20. 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
for
and 0 otherwise

21. Graphs of standard beta probability density function

22. Beta Distribution – Properties
If X ~ B( , A, B),
then
Mean or expected value
Standard deviation

23. 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 (in
days) necessary for laying the foundation has a beta
distribution with A = 2, B = 5, α = 2, and β = 3. Then

24. Beta Distribution – Example (continue)
, so For these values of α
and β, the probability density functions of is a simple
polynomial function. The probability that it takes at most
3 days to lay the foundation is