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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 25 x 0.407 . 4 2 4 11 11 27 2 27 4 27 24

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