Definition  5.7.2 Gamma Distributions. Let α and β be positive numbers. A random variable X has the gamma distribution with parameters α and β if X has a continuous distribution for which the p.d.f. is

Theorem 5.7.5 Moments. Let X have the gamma distribution with parameters α and β. For k = 1, 2,..., 

In particular, E(X) = α/β , and Var(X) = α/β^2 .

Sum 

Theorem 5.7.7 If the random variables X1,...,Xk are independent, and if Xi has the gamma distribution with parameters αi and β (i = 1, . . . , k), then the sum X1 + ... + Xk has the gamma distribution with parameters α1 + ... + αk and β.

Exponential Distribution

Definition 5.7.3 Exponential Distributions. Let β > 0. A random variable X has the exponential distribution with parameter β if X has a continuous distribution with the p.d.f.

Theorem 5.7.8 The exponential distribution with parameter β is the same as the gamma distribution with parameters α = 1 and β. If X has the exponential distribution with parameter β, then

Theorem 5.7.9 Memoryless Property of Exponential Distributions. Let X have the exponential distribution with parameter β, and let t > 0. Then for every number h > 0,