Download (c) Compute E(log(z)ly).(5 pts)

1. (20 pts] Consider the following stochastic formulation
X
XrvN(0,(T2),
Z
rvr(r/2,r/2),
X1..Z
HU
y= y'Z'
where r(a,,B) stands for a gamma distribution with mean a/,B and '1..' indi­
cates independence. Obtain
(a) The marginal p.d.f, of Y, i.e., j(y) =7 (5 pts)
(b) The conditional p.d.f. of ZIY = y, say, j(zly) =7 (5 pts)
(c) Compute E(log(z)ly).(5 pts)
(d) Derive the marginal distribution of W = y 2/ (T2. (5 pts)
2. (15 pts) Suppose that if 0
=
1, then y has a normal distribution with mean
1 and standard deviation (T, and if 0 = 2, the y has a normal distribution
NC
with mean 2 and standard deviation (T. Also, suppose P(O
= 1) =
0.5 and
P(O = 2) = 0.5.
(a) For (T = 2, write the formula for the marginal probability density for y7
(b) What is P(Oly
= 1), again supposing (T = 27
(c) Describe how the posterior density of 0 changes in shape as (T is increased
and as it is decreased.
3. (15 pts) Write (in English) the purposes (5 pts) and plans (10 pts) for studying
a PHD in Statistics.
4. (16 pts) One observation, X, is taken from a N(O, (T2) population. Find the
MLE (8 pts) and the method of moments estimator (8 pts) of (T .
5. (14 pts)IfS2 is the sample variance based on a sample of size n from a normal
population, we know that (n - 1)8 2 j a 2 has a X;-l distribution. The con­
=
HU
jugate prior for a 2 is the inverted gamma pdf, IG(O'., (3), given by 1r(a2 )
2)
e- 1/ ([3oc j[r(a)(3OC(a2)OC+l], a 2 > 0, where
0'.
and (3 are positive constants. Find
the posterior distribution of a 2 (10 pts) and the posterior mean ofthis distri­
bution (4 pts).
6. (20 pts) Let X be a single observation from a distribution with pdf j(xIO)
0--:1
Ox· ,0 < x < 1, 0 < 0 <
=
00.
(a) Find the most powerful test of lfo : 0 = 00 vs HI :0 = 0ll 01 > 00 . (8
pts)
(b) Find the size a critical region. (6 pts)
NC
(c) Find the power of the most powerful test. (6 pts)