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 . !