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.