ABSTRACT
Consider an outcome x that follows a gamma distribution with parameters α and n where n is an integer. The function that governs its probabilities is
0( ) . n
n x xf x x en
In this section, we will assume that n itself has a negative binomial distribution. The probability of any value of n, which we write as ,nP is
1 1
n P p p
r −−⎛ ⎞= −⎜ ⎟−⎝ ⎠
for n = r, r + 1, r + 2, r + 3, … , ∞. We are interested in finding the marginal distribution of x. Using the Law of Total Probability we write
( ) |
1 1
11 1 . 11
f x f x n P
n x e p p
rn
npe x p rx p n
α
α
=
−⎛ ⎞ = −⎜ ⎟−Γ ⎝ ⎠
−⎛ ⎞ ⎛ ⎞ = −⎜ ⎟ ⎜ ⎟−− Γ ⎝ ⎠⎝ ⎠
(F.1)
Recognizing that
1 ( ) 1 ( )( )!
