. By working with the function H(s) = GN (1 − s) or otherwise, deduce that N has a Poisson distribution.

You may assume that



1 + (x/n) + o(n−1)

n

→ex as n → ∞. (Cambridge 2002)

Example 5.22 If X has the exponential distribution with parameter λ, then FX (x) = (
0 if x ≤ 0, 1 − e−λx if x > 0, with density function fX (x) = (
0 if x ≤ 0, λe−λx if x > 0. △