STAT 410
Homework #9
Spring 2008
(due Monday, March 31, by 4:30 p.m.)
No credit will be given without supporting work.
Warmup:
4.2.3
By Chebyshev’s Inequality,
P
(

W
n
–
μ

≥
ε
)
≤
2
2
W
°
±
n
=
2
°
p
n
b
→
0 as
n
→
∞
for all
ε
> 0.
Therefore,
²
W
P
n
→
.
1.
4.3.2
F
Y
1
(
x
)
=
(
)
θ



x
n
e
1
,
x
>
θ
.
(
Recall Homework 8, problems 6 & 7.
)
F
Z
n
(
z
)
= P
(
n
(
Y
1
–
θ
)
≤
z
)
= P
(
Y
1
≤
n
z
+
θ
)
= 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.
)
2.
4.3.3
F
Y
n
(
x
)
=
(
)
(
)
n
x
F
.
Since Z
n
=
n
(
1 – F
(
Y
n
)
)
, P
(
Z
n
≥
0
)
= 1.
Let
z
> 0.
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F
Z
n
(
z
)
= P
(
n
(
1 – F
(
Y
n
)
)
≤
z
)
= P
(
F
(
Y
n
)
≥
1 –
n
z
)
=
°
°
±
²
³
³
´
µ
°
±
²
³
´
µ



n
z
n
1
F
F
1
1
Y
=
n
n
z
1
1
°
±
²
³
´
µ


→
1 –
e
–
z
as
n
→
∞
.
For the limiting distribution Z of Z
n
,
F
Z
(
z
)
= 1 –
e
–
z
,
z
> 0,
f
Z
(
z
)
=
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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 Spring '08
 AlexeiStepanov
 Probability

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