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

Gamma Distribution

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

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.

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

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 ³ G - x for e b a otherwise,

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

Gamma density functions

Standard gamma density functions

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. ~

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

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

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

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.

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

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

Beta Distribution

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

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

Graphs of standard beta probability density function

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

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

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