Homework 4 Solutions
8.62
We have f (x|) = ex ; x > 0, > 0 with x = 5.1, n = 20. The prior is
1 e .
h() =
()
Therefore, the posterior is
f (|x1 , . . . , xn ) f (x1 , . . . , xn |)h() n+ e(+
xi )
.
We see that this is proportional to the gamma p.d.f. with
Homework 5 Solutions
8.6
b. The Cramr-Rao inequality is
e
V (T )
1
n I()
[here n = 1]
where (from homework 2)
I(p) = E( (p, X) = E
X
nX
2
p
(1 p)2
=
n
E(X) n E(X)
+
=
.
2
2
p
(1 p)
p(1 p)
Therefore,
p(1 p)
.
n
From homework 2, the m.l.e. of p is p = X s
Associate Professor J. Kerr
Midterm I
STAT 6501
Fall 2015
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your bluebook starting each new problem on a new page, including all pertinent work for
full credit. Partial c
Homework 3 Solutions
1. Find the m.l.e. of based on a random sample X1 , X2 , . . . , Xn from
each of the following p.d.f.s.
(a)
f (x|) = x1 ,
0 < x < 1, 0 <
n
() = n ln() + ( 1)
ln(xi )
i=1
d ()
n
= +
d
and
ln(xi )
i=1
d2 ()
n
= 2
d2
So
n
ln(xi )
=
and
Associate Professor J. Kerr
Midterm II
STAT 6501
Fall 2015
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your bluebook starting each new problem on a new page, including all pertinent work for
full credit. Partial
Homework 2 Solutions
8.6
a. Since f (x) =
n
x
px (1 p)nx , the log-likelihood function is
(p) = ln
n
+ x ln(p) + (n x) ln(1 p).
x
The rst derivative w.r.t. p is
d (p)
x nx
=
dp
p
1p
1
x
which, when set to zero, yields p 1 = n 1 p = n .
x
The second deri
Associate Professor J. Kerr
Final Exam
STAT 6501
Fall 2015
Instructions: You have 110 minutes to complete the exam. Please write all solutions in
your greenbook starting each new problem on a new page, including all pertinent work for
full credit. Partial