STAT 410
Homework #10
Fall 2009
(due Friday, November 6, by 3:00 p.m.)
Warmup:
4.3.7
Hint:
( 29
( 29
α
β
1
1
M
Gamma
t
t

=
,
t
<
1
/
.
M
Y
n
(
t
)
=
E
(
e
t
X
n
/
n
)
=
M
X
n
(
t
/
n
)
=
n
n
t
1
1

→
e
t
as
n
→
∞
.
Let X be a “random” variable with P
(
X =
) = 1.
(
degenerate distribution at
)
Then
M
X
(
t
)
=
E
(
e
t
X
)
=
e
t
.
⇒
X
Y
D
n
→
.
1.
4.3.2
F
Y
1
(
x
)
=
( 29
θ



x
n
e
1
,
x
>
.
(
)
F
Z
n
(
z
)
=
P
(
n
(
Y
1
–
)
≤
z
)
=
P
(
Y
1
≤
n
z
+
)
=
1 –
e
–
z
,
z
> 0.
Therefore,
the limiting distribution of
Z
n
is Exponential with mean 1.
(
Exponential distribution with mean 1 is same as Gamma distribution with
α
= 1,
β
= 1.
)
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4.3.4
Hint:
F
Y
2
(
x
)
=
( 29
( 29 ( 29
( 29
∑
=


⋅
⋅
n
i
i
n
i
x
x
i
n
2
F
1
F
=
( 29
( 29 ( 29
( 29 ( 29
( 29
1
F
1
F
F
1
1





⋅
⋅
n
n
x
x
n
x
.
F
Y
2
(
x
)
=
( 29
( 29 ( 29
( 29 ( 29
( 29
1
F
1
F
F
1
1





⋅
⋅
n
n
x
x
n
x
.
Since
W
n
=
n
F
(
Y
2
),
P
(
0 < W
n
<
n
) = 1.
Let
0 <
w
<
n
.
F
W
n
(
w
)
=
P
(
n
F
(
Y
2
)
≤
w
)
=

n
w
1
Y
F
F
2
=
1
1
1
1





⋅
⋅
n
n
n
w
n
w
n
n
w
→
1 –
e
–
w
–
w
e
–
w
as
n
→
∞
.
For the limiting distribution
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 Fall '08
 Monrad

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