8-1. Show that each of the following is an exponential family. Identify the
natural parameter and natural statistic.
(a) The Poi() family of distributions.
(b) The Exp() family of distributions.
(c) The Gam(
; ) family of distributions with both parameters unknown.
The natural parameter vector and natural statistic vector are both two-
dimensional.
8-2. Suppose X is Poi() and the prior distribution for is Gam(
; ),
where
and are hyperparameters. Find the posterior distribution for .
8-3. Suppose X1, : : :, Xn are IID Gam(
; ), where
is known and is
unknown. Suppose the prior distribution for is Gam(
0; 0), where
0
and 0 are hyperparameters. Find the posterior distribution for .
8-4. Suppose X1, : : :, Xn are IID Unif(0; ) and the prior distribution for
is Unif(a; b), where a and b are hyperparameters. Find the PDF of the
posterior distribution for . Under what conditions on x1, : : :, xn, a, and b
does the solution make no sense?
8-5. Suppose the distribution for data X is Geo(p). Show that the beta
family of distributions is conjugate.
8-6. Suppose X1, : : :, Xn are IID N(; 1=), where is known and is
unknown. Find a brand-name family of distributions that is conjugate.
8-7. Suppose X is Geo(p) and the prior distribution for p is Beta(
1;
2),
where
1 and
2 are hyperparameters. Find the posterior distribution for
p.
8-8. Suppose X1, : : :, Xn are IID N(; 1=), where is known and
is unknown. Suppose the prior distribution for is a distribution in the
brand-name conjugate family of distributions found in problem 8-6. Find
the posterior distribution for .
1
8-9. Suppose the situation is the same as in problem 8-8. Find the poste-
rior distribution for =
p
1=. Hint: change-of-variable formula.
8-10. Suppose X1, : : :, Xn are IID Exp().
(a) Suppose the prior distribution for is
at (an improper prior). Find
the posterior distribution for .
(b) Suppose the prior distribution for is proportional to ?1 (an improper
prior). Find the posterior distribution for .
Review Problems from Previous Tests
8-11. Suppose X1, : : :, Xn are IID Exp(), and suppose the prior distri-
bution for is Gam(
0; 0), where
0 and 0 are hyperparameters. Find
the posterior distribution for :
8-12. Suppose X is Poi(). We have only one observation. And suppose
the prior distribution for is proportional to ?1=2, an improper prior.
(a) Find the posterior distribution for .
(b) For what values of the data x does your answer to part (a) make sense?
