STAT 410
Homework #8
Summer 2009
(due Wednesday, July 22, by 4:00 p.m.)
Warmup:
4.3.7
Hint:
( 29
(
29
α
β
1
1
M
Gamma
t
t

=
,
t
<
1
/
β
.
1.
4.3.2
2.
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
.
3.
a)
4.3.9
Hint:
We already know that
(
29
(
29
1
,
0
2
Y
N
D
n
n
n
→

.
b)
Find
P
(
40 < X < 60
)
,
where
X
has a
(
29
50
2
χ
distribution.
Hint:
Use integration by parts 24 times
or
EXCEL:
=
CHIINV
(
α
,
v
)
gives
( 29
v
2
α
χ
for
2
χ
distribution with
v
degrees
of freedom,
x
s.t.
P
(
( 29
v
2
χ
>
x
)
=
α
.
=
CHIDIST
(
x
,
v
)
gives
the upper tail probability for
2
χ
distribution
with
v
degrees of freedom,
P
(
( 29
v
2
χ
>
x
)
.
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4.
4.3.18
5.
Let
λ
> 0
be an unknown parameter and let
X
1
, X
2
, … , X
n
be independent
random variables, each with the probability density function
f
(
x
)
=
(
29
<
<


otherwise
0
1
0
1
1
λ
λ
x
x
.
a)
Obtain the maximum likelihood estimator of
λ
,
n
λ
ˆ
.
b)
Find the CDF of
X
1
.
(
not an order statistic
)
c)
Let
W
i
= –
ln
(
1 – X
i
)
,
i
= 1, 2, … ,
n
.
Find the CDF and the PDF of
W
1
.
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 Spring '08
 STEPANOV
 Statistics, Normal Distribution, Probability theory, probability density function, Cumulative distribution function, CDF, λ

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