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Special Continuous Probability Distributions Gamma Distribution Beta Distribution

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## Special Continuous Probability Distributions Gamma Distribution Beta Distribution

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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**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**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**for**Standard Gamma Distribution The standard gamma distribution has The probability density function of the standard Gamma distribution is: And is 0 otherwise**~**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.**, then**If x~ G Gamma Distribution - Properties • 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.**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**Example - continue**The probability that a mouse survives at least 30 weeks is**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**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**If X ~ B( , A, B),**Beta Distribution – Properties 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