Special Continuous Probability Distributions Gamma Distribution Beta Distribution

# Special Continuous Probability Distributions Gamma Distribution Beta Distribution

## Special Continuous Probability Distributions Gamma Distribution Beta Distribution

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1. 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. x 1 - a - b ³ 1 x e for x 0 , a b G a ( ) 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. otherwise, 0

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

8. Gamma density functions

9. Standard gamma density functions

10. ~ Probability Distribution Function 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.

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

12. 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

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

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

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

17. Beta Distribution

18. a b f ( x ; , , A , B ) a - b - 1 1 G a + b - - æ ö æ ö 1 ( ) x A B x = × ç ÷ ç ÷ , - G a × G b - - B A ( ) ( ) B A B A è ø è ø £ £ for A x B and is 0 otherwise, Beta Distribution - Definition A random variable is said to have a beta distribution with parameters, , , and if the probability density function of is where

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 for and 0 otherwise

20. Graphs of standard beta probability density function

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

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 (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) , 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