1. 1 Special Continuous Probability Distributions Gamma Distribution Beta Distribution Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08

2. 2 Gamma Distribution

3. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 3 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 The Gamma Distribution

4. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 4 Definition For, the gamma function is defined by Properties of the gamma function: 1.For any [via integration by parts] 2. For any positive integer, 3. Gamma Function

5. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 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. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 6 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.,0 )( 1 1     xforex x     otherwise, 0 Gamma Distribution - Definition

7. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 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. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 8 Gamma density functions

9. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 9 Standard gamma density functions

10. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 10 If 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. ~ Probability Distribution Function

11. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 11 Mean or Expected Value Standard Deviation Gamma Distribution - Properties If x ~ G, then

12. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 12 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 Gamma Distribution - Example

13. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 13 The probability that the reaction time is more than 4 sec is Gamma Distribution – Example (continued) Where and

14. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 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 Where is the incomplete gamma function. MINTAB and other statistical packages will calculate once values of x,, and have been specified.

15. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 15 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 Example

16. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 16 The probability that a mouse survives at least 30 weeks is Example - continue

17. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 17 Beta Distribution

18. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 18 Beta Distribution - Definition A random variable is said to have a beta distribution with parameters,,, and if the probability density function of is 0, )()( )(1 ),,,;( 11 otherwise,isand BxAfor AB xB AB Ax AB BAxf                           where

19. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 19 Standard Beta Distribution for and 0 otherwise If X ~ B(, A, B), A =0 and B=1, then X is said to have a standard beta distribution with probability density function

20. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 20 Graphs of standard beta probability density function

21. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 21 Beta Distribution – Properties If X ~ B(, A, B), then Mean or expected value Standard deviation

22. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 22 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 Beta Distribution – Example

23. Stracener_EMIS 7370/STAT 5340_Sum 08_06.19.08 23, 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 Beta Distribution – Example (continue)