4.6 The Gamma Probability Distribution

116 Chapter 4. Continuous Variables and Their Probability Distributions (ATTENDANCE 7) The Gamma Probability DistributionThe continuousgammarandom variableYhas densityf(y) ={y 1e y/ ( ),0 y < ,0,elsewhere,where the gammafunctionis defined as ( ) = 0y 1e ydyand its expected value (mean), variance and standard deviation are, =E(Y) = , 2=V(Y) = 2, = V(Y).One important special case of the Gamma , is the continuouschi squarerandom vari-ableYwhere = 2and = 2; in other words, with densityf(y) ={y 22e y22 /2 ( 2),0 y < ,0,elsewhere,and its expected value (mean), variance and standard deviation are, =E(Y) = , 2=V(Y) = 2 , = V(Y).Another important special case of the Gamma , is the continuousexponentialrandomvariableYwher e = 1; in other words, with densityf(y) ={1 e y/ ,0 y < ,0,elsewhere,and its expected value (mean), variance and standard deviation are, =E(Y) = , 2=V(Y) = 2, =.}}}

Exercise 4.6 (The Gamma Probability Distribution) 1. Gamma distribution. (a) Gamma function8, Γ(α). 8The gamma functionis a part of the gamma density. There is no closed–form expression for the gamma function except when α is an integer. Consequently, numerical integration is required. We will mostly use the calculator to do this integration.

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