ABSTRACT
Consider an outcome x that follows a gamma distribution with parameters α and m where m is an integer greater than zero. The function that governs its probabilities is
0( ) . m
m x xf x x em
In this situation the parameter λ has its own probability distribution
0( ) , r
rf e r
where we will assume that r is also a positive integer. We are interested in finding the marginal distribution of x. Using the Law of Total Probability we write
( ) ( | ) ( )
f x f x f
x e e d m r
λ λ
λ α λ λ ∞
=
= Γ Γ
∫ (G.1)
Removing terms involving λ, the second line of expression (F.1) becomes
( ) . r
xm r mf x x e d m r
Recognizing that the integrand in expression (G.2) is related to that of a variable that follows a gamma distribution, we include the appropriate constant so that this integral’s value is one.