This preview shows pages 1–3. Sign up to view the full content.
STAT 409
Fall 2011
Version A
Name
ANSWERS
.
Exam 1
Page
Earned
/
70
Be sure to show all your work; your
partial credit might depend on it.
Put your final answers at the end of
your work, and mark them clearly.
No credit will be given
without supporting work.
The exam is closed book and closed notes.
You are allowed to use a calculator and
one 8½" x 11" sheet with notes.
1
2
3
4
5
6
Total
___________________________________________________________________________
Academic Integrity
The University statement on your obligation to maintain academic integrity is:
If you engage in an act of academic dishonesty, you become liable to severe disciplinary action. Such acts include cheating;
falsification or invention of information or citation in an academic endeavor; helping or attempting to help others commit
academic infractions; plagiarism; offering bribes, favors, or threats; academic interference; computer related infractions; and
failure to comply with research regulations.
Rule 33 of the Code of Policies and Regulations Applying to All Students gives complete details of rules governing
academic integrity for all students. You are responsible for knowing and abiding by these rules.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 1.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
( ) ( ) ( )
α
1
1
α
α
;
X
x
x
f

+
=
⋅
,
0 <
x
< 1,
α
> – 1.
a)
(4)
Find a sufficient statistic
Y =
u
(
X
1
, X
2
, … , X
n
)
for
α
.
θ
=
α
( )
∏
=
n
i
i
f
1
X
θ
X
;
=
( ) ( ) ( ) ( )
θ
θ
1
1
X
1
1
X
1
1
θ
θ

+

+
∏
⋅
⋅
=
=
=
∏
n
i
i
n
n
i
i
.
By Factorization Theorem,
Y
1
=
( )
∏
=

n
i
i
1
X
1
is a sufficient statistic for
θ
.
⇒
Y
2
=
ln
Y
1
=
( )
∑
=

n
i
i
1
X
1
ln
is also a sufficient statistic for
θ
.
OR
f
(
x
;
θ
)
=
( ) ( )
{ }
1
1
θ
θ
ln
ln
exp
+
+

⋅
x
.
⇒
K
(
x
) =
( )
1
ln
x

.
⇒
Y
2
=
( )
∑
=
n
i
i
1
X
K
=
( )
∑
=

n
i
i
1
X
1
ln
is a sufficient statistic for
θ
.
⇒
Y
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/13/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.
 Fall '11
 STEPHANOV

Click to edit the document details