Stats Help

no explanation. If the problem is a proof, then you need words as well as

formulas. Explain why your formulas follow one from another.

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?

2