STAT 409
Homework #3
Fall 2011
( due Friday, September 16, by 4:00 p.m. )
1.
a)
Let
X
have a
χ
2
(
r
)
distribution.
If
k
> –
r
/
2
,
show that
E
(
X
k
)
exists
and it is given by
E
(
X
k
)
=
Γ
+
Γ
2
2
2
r
k
r
k
.
E
(
X
k
)
=
( )
( )
∫
∞


Γ
⋅
0
2
1
2
2
2
2
1
dx
e
x
r
x
x
r
r
k
=
∫
∞

+
+

+
Γ
Γ
+
Γ
⋅
0
2
1
2
2
2
2
1
2
2
2
dx
e
x
k
r
r
k
r
x
k
r
k
r
k
=
Γ
+
Γ
2
2
2
r
k
r
k
,
since
2
1
2
2
2
2
1
x
k
r
k
r
e
x
k
r

+
Γ

+
+
is the
p.d.f. of
χ
2
(
r
+ 2
k
)
distribution.
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6.114 (a), (b)
(
)
a)
Recall that
(
n
– 1
) S
2
/
σ
2
has a
χ
2
(
n
– 1
)
distribution.
From part (a), if
r
=
n
– 1
and
k
=
1
/
2
,
then

Γ
Γ

=
2
1
2
2
S
1
E
σ
n
n
n
.
Therefore,
σ
S
2
2
2
1
1
E
=
⋅
Γ

Γ

n
n
n
,
and
S
2
2
2
1
1
⋅
Γ

Γ

n
n
n
is unbiased for
σ
.
b)
n
= 5
c
=
Γ
Γ
2
5
2
2
4
4
=
π
2
1
2
3
2
1
2
⋅
⋅
=
π
2
3
8
≈
1.063846.
n
= 6
c
=
Γ
Γ
2
6
2
2
5
5
=
2
2
2
1
2
3
5
π
⋅
⋅
=
2
8
5
3
π
≈
1.050936.
2.
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 Fall '11
 STEPHANOV
 Normal Distribution, 2 K, 2k, 8 degrees, 90%

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