1.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
( )
4
3
θ
4
θ
x
e
x
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
3
1
4
θ
4
θ
.
By Factorization Theorem,
Y =
∑
=
n
i
i
1
4
X
is a sufficient statistic for
θ
.
OR
f
(
x
;
θ
)
=
{
}
4
ln
ln
ln
exp
3
4
θ
θ
x
x
+
+
+

⋅
.
⇒
K
(
x
)
=
x
4
.
⇒
Y
=
(
)
∑
=
n
i
i
1
X
K
=
∑
=
n
i
i
1
4
X
is a sufficient statistic for
θ
.
b)
What is the probability distribution of
Y =
∑
=
n
i
i
1
4
X
?
F
X
(
x
)
=
4
θ
1
x
e


,
x
> 0
θ
> 0.
F
W
(
w
)
=
P
(
X
4
≤
w
)
=
w
e
θ
1


,
w
> 0
θ
> 0.
W = X
4
has
Exponential
distribution with
mean = “usual
θ
” =
θ
1
.
Y =
∑
=
n
i
i
1
4
X
=
∑
=
n
i
i
1
W
has
Gamma
distribution with
α
=
n
and
“usual
θ
” =
θ
1
.
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c)
Use part (b) to suggest a confidence interval for
θ
with
(
1

α
)
100 %
confidence level.
2
W
/
“usual
θ
”
=
2
θ
∑
=
n
i
i
1
4
X
has a
chisquare
distribution with
r
=
2
α
=
2
n
d.f.
⇒
P
(
(
)
2
2
1
2
χ
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 Fall '11
 STEPHANOV
 Normal Distribution, Probability, Sufficient statistic

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