Recall that a probability distribution belongs to the exponential family if its distribu-
tion function takes the following form:
f (yi, θi, φ) = exp
yiθi − b(θi)
a(φ) + c(yi, φ)
, (1)
where θi is the natural location parameter and φ is the scale parameter.
(a) Show that the Poisson distribution belongs to the exponential family. Comparing
with (1), you should show that θi = log μi, b(θi) = μi, a(φ) = 1.
(b) Also, since we know that for exponential families, E(Y ) = b′(θi) and V ar(Y ) =
a(φ)b′′(θi), obtain the expected value and variance for the Poisson distribution.
As a reminder, the probability mass function of a Poisson distribution is
f (yi; μi) = exp(−μi)μyi
i
yi!
for yi ∈ {0, 1, 2, 3, · · ·