ABSTRACT
Distribution A first simple example of “compounding” combines the binomial distribution with the Poisson distribution. Consider an outcome X that follows a binomial distribution where n and θ are fixed. In this example θ will remain a constant. We write the [ ]|X k n=P as
[ ] ( )| 1 .n kknX k n k
θ θ − ⎛ ⎞
= = −⎜ ⎟ ⎝ ⎠
P (A.1)
However, assume that n is not constant, but follows a Poisson distribution with parameter λ. We are interested in computing the unconditional distribution of X. Using the law of total probability, we write
| ,X k X k n n ∞
= =∑ n=
P = P P
|
1 . !