3.2 If the random variable Y has the Gamma distribution with a scale parameter , which is the parameter of interest, and a known shape

3.2 If the random variable Y has the Gamma distribution with a scale parameter β, which is the parameter of interest, and a known shape parameter

α, then its probability density function is f(y;β) = β

α

Γ(α)

y

α−1 e

−yβ

.

Show that this distribution belongs to the exponential family and find the natural parameter. Also using results in this chapter, find E(Y) and var(Y).