5.2The Poisson Random Variable 149

0

.20

.16

.12

.08

.04

(^012345678910)
P {X = i}
11 12 i
FIGURE 5.3 The Poisson probability mass function withλ= 4.
As a prelude to determining the mean and variance of a Poisson random variable, let
us first determine its moment generating function.
φ(t)=E[etX]

∑∞
i= 0
etie−λλi/i!
=e−λ
∑∞
i= 0
(λet)i/i!
=e−λeλe
t
=exp{λ(et−1)}
Differentiation yields
φ′(t)=λetexp{λ(et−1)}
φ′′(t)=(λet)^2 exp{λ(et−1)}+λetexp{λ(et−1)}