Stat 5102 (Geyer) Spring 2010
Homework Assignment 8
Due Wednesday, March 31, 2010
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
81.
Show that each of the following is an exponential family. Identify the
natural parameter and natural statistic.
(a) The Poi(
μ
) family of distributions.
(b) The Exp(
λ
) family of distributions.
(c) The Gam(
α, λ
) family of distributions with both parameters unknown.
The natural parameter vector and natural statistic vector are both two
dimensional.
82.
Suppose
X
is Poi(
μ
) and the prior distribution for
μ
is Gam(
α, λ
),
where
α
and
λ
are hyperparameters. Find the posterior distribution for
μ
.
83.
Suppose
X
1
,
. . .
,
X
n
are IID Gam(
α, λ
), where
α
is known and
λ
is
unknown.
Suppose the prior distribution for
λ
is Gam(
α
0
, λ
0
), where
α
0
and
λ
0
are hyperparameters. Find the posterior distribution for
λ
.
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 Spring '03
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 λ, posterior distribution, exponential family, prior distribution

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