STAT 409
Homework #3
Fall 2009
(due Friday, September 18, by 4:00 p.m.)
1.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
from the distribution with
probability density function
( 29
(
29
(
29
θ
2
X
X
ln
1
θ
θ
;
x
x
x
f
x
f
⋅
-
=
=
,
x
> 1,
θ
> 1.
a)
Find the sufficient statistic
Y =
u
(
X
1
, X
2
, … , X
n
)
for
θ
.
f
(
x
1
;
θ
)
f
(
x
2
;
θ
)
…
f
(
x
n
;
θ
)
=
(
29
∏
=
⋅
-
n
i
i
i
x
x
1
θ
2
ln
1
θ
=
(
29
∏
∏
=
-
=
⋅
⋅
-
n
i
i
n
i
i
n
x
x
1
θ
1
2
ln
1
θ
.
⇒
Y
1
=
∏
=
n
i
i
1
X
is a sufficient statistic for
θ
.
⇒
Y
2
=
ln
Y
1
=
ln
∏
=
n
i
i
1
X
=
∑
=
n
i
i
1
X
ln
is also a sufficient statistic for
θ
.
OR
(
29
(
29
θ
2
X
ln
1
θ
θ
;
x
x
x
f
⋅
-
=
=
exp
{
–
θ
ln
x
+
ln
ln
x
+ 2
ln
(
θ
– 1
)
}
⇒
K
(
x
)
=
ln
x
.
⇒
Y
2
=
∑
=
n
i
i
1
X
ln
is a sufficient statistic for
θ
.
⇒
Y
1
=
e
Y
2
=
∏
=
n
i
i
1
X
is also a sufficient statistic for
θ
.
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b)
What is the probability distribution of
W =
ln
X
?
Using the change-of-variable technique
(
Section 3.5, pp. 173 – 174
)
:
X
–
continuous
r.v.
with
p.d.f.
f
X
(
x
)
.
Y
=
u
(
X
)
u
(
x
)
–
one-to-one, differentiable
(
strictly increasing or strictly decreasing
)
X
=
u
–
1
(
Y
)
=
v
(
Y
)
⇒
f
Y
(
y
)
=
f
X
(
v
(
y
)
)
⋅
|
v
'
(
y
)
|
W =
ln
X
X =
e
W
=
v
(
W
)
v
'
(
w
)
=
e
w
f
W
(
w
)
=
(
θ
– 1
)
2
w
e
–
w
θ
⋅
e
w
=
(
θ
– 1
)
2
w
e
–
w
(
θ
– 1
)
=
(
(
29
2
2
1
θ
Γ
-
w
2 – 1
e
–
w
(
θ
– 1
)
,
w
> 0.
⇒
W
has
Gamma
(
α
= 2, “usual
θ
” =
1
1
θ
-
)
distribution.
c)
What is the probability distribution of
∑
=
n
i
i
1
X
ln
?
Suppose
X
and
Y
are
independent
,
X
is
Gamma
(
α
1
,
θ
)
,
Y
is
Gamma
(
α
2
,
θ
)
.
If random variables X and Y are independent, then M
X + Y
(
t
)
=
M
X
(
t
)
⋅
M
Y
(
t
)
.
⇒
M
X + Y
(
t
)
=
(
29
(
29
2
1
α
α
θ
θ
1
1
1
1
t
t
-
-
⋅
=
(
29
2
1
α
α
θ
1
1
+
-
t
,
t
<
θ
1
.
⇒
X + Y
is
Gamma
(
α
1
+
α
2
,
θ
)
;
⇒
∑
=
n
i
i
1
X
ln
=
∑
=
n
i
i
1
W
has
Gamma
(
α
= 2
n
, “usual
θ
” =
1
1
θ
-
)
distribution.
2.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
( 29
(
29
X
X
θ
2
θ
θ
;
x
e
x
x
f
x
f
-
=
=
x
> 0
θ
> 0.
a)
Find the sufficient statistic
Y =
u
(
X
1
, X
2
, … , X
n
)
for
θ
.
f
(
x
1
,
x
2
,
…
x
n
;
θ
)
=
f
(
x
1
;
θ
)
f
(
x
2
;
θ
)
…
f
(
x
n
;
θ
)
=
∏
=
∑
=
-
n
i
i
x
n
n
x
n
i
i
e
1
1
1
2
θ
θ
.
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- Spring '11
- Stepanov
- Normal Distribution, Probability, Variance, Probability theory, probability density function, Sufficient statistic
-
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