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STAT 410
Summer 2009
Homework #6
(due Wednesday, July 15, by 4:00 p.m.)
1.
Let
X
be a continuous random variable with probability density function
( 29
β
1
X
α
β
β
α
x
e
x
x
f


=
,
x
> 0,
where
α
> 0,
β
> 0.
(
X has a Weibull distribution.
)
Consider
Y =
β
X
.
What is the probability
distribution of
Y?
2.
Let
X
have an exponential distribution with
θ
= 1;
that is, the
p.d.f. of
X
is
f
(
x
) =
e
–
x
,
0 <
x
<
∞
.
Let
T
be defined by
T =
ln
X.
a)
Show that the
p.d.f. of
T
is
g
(
t
) =
e
t
e
–
e
t
,
–
∞
<
t
<
∞
,
which is the
p.d.f. of an extreme value distribution.
b)
Let
W
be defined by
T =
α
+
β
ln
W,
where
–
∞
<
α
<
∞
and
β
> 0.
Show that
W
has a Weibull distribution.
3.
3.4.11
Let the random variable
X
have the
p.d.f.
f
(
x
)
=
2
2
2
2
x
e

π
,
0 <
x
<
∞
,
zero elsewhere.
Find the mean and the variance of
X.
Hint:
Compute
E
(
X
)
directly and
E
(
X
2
)
by comparing the integral with the integral
representing the variance of a random variable that is
N
(
0, 1
).
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View Full Document4.
3.4.10
If
e
3
t
+ 8
t
2
is the
mgf
of the random variable
X,
find
P
(
–
1 < X < 9
).
5.
3.4.28
Let
X
1
and
X
2
be independent with normal distributions
N
(
6, 1
)
and
N
(
7, 1
),
respectively.
Find
P
(
X
1
> X
2
).
Hint:
Write
P
(
X
1
> X
2
) = P
(
X
1
– X
2
> 0
)
and determine the distribution
of
X
1
– X
2
.
6.
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 Spring '08
 STEPANOV
 Statistics, Probability

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