STAT608 Mathematical Statistics II Homework 7
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STAT608 Mathematical Statistics II Homework 7
Please show your work in detail to get the full credits.
1. Exercise Problem 7.24 (a), (b).
(c) Describe how to construct a (1 _ a) equal-tail posterior interval for 9 .
(d) Describe how to construct a (1 _ a) highest posterior density (HPD) interval for 9 .
2. Exercise Problem 9.26.
3. Exercise Problem 9.27 (a), (b), (c).
4. Exercise Problem 9.33 (a), (b).
5. Let y1 , y2 , . . . , yn be a random sample from Bernoulli(9) and define x = 
yi . Thus, xl9 ~
Binomial(9) and the sampling density for x is
f(sl9) = ╱ 、s(n) 9α (1 _ 9)n −α for s = 0, 1, . . . n
where 0 < 9 < 1 and n is any positive integer. Let the prior distribution for the parameter 9 be the Beta distribution with the following probability density function:
f(9la, 8) = 
9a − 1 (1 _ 9)8 − 1
where 0 < 9 < 1, a 义 0 and 8 义 0. Note that for 9 ~ Beta(a, 8),
E(9) = and Var(9) =
(a) Find the posterior distribution of 9 .
(b) Find the posterior mean of 9 .
(c) Describe how to construct a 90% equal-tail posterior interval for 9 .
6. Let x1 , x2 , . . . , xn be a random sample from a Normal(u, 1) distribution:
f(slu, 1) = 
exp , 
}.
(a) Prove that
exp , _ i
1 (si _ u)2 } = exp , _ (n _ 1)52 _ n(
_ u)2 、
where 52 = 
(si _ 
)2 .
(b) Let T = 
xi . Show that T is sufficient using the likelihood of u.
(c) Let u has a Normal (9, r2 ) prior distribution. Derive the posterior distribution of u. (d) Find the posterior mean of u.
(e) Find a 95% equal-tail posterior interval for u.
7. Let x1 , x2 , . . . , xn be a random sample from Poisson(9):
exp(_9) 9α
f (sl9) = s! , 9 义 0 and s = 0, 1, 2, . . . .
We found in an earlier assignment that the maximum likelihood estimator for 9 is 9ˆMLE = 
=
1
n
xi . Consider the prior distribution for the
parameter
9 as a Gamma distribution:
f (9la, 8) = 
9a − 1 exp(_9/8), 9 义 0, a 义 0 and 8 义 0.
Note that for 9 ~ Gamma(a, 8),
E(9) = a8 and Var(9) = a82 .
(a) Find the posterior distribution of 9 .
(b) Find the posterior mean of 9 .
(c) Describe how to construct a (1 _ a) equal-tail posterior interval for 9 .
(d) Describe how to construct a (1 _ a) highest posterior density (HPD) interval for 9 .
2023-05-24