STAT 410
Homework #7
Summer 2009
(due Monday, July 20, by 4:00 p.m.)
Warmup:
4.2.3
1.
a)
4.1.4
b)
4.1.8
a)
4.1.4
Let
X
1
, X
2
, X
3
, X
4
be four iid random variables having the same pdf
f
(
x
)
= 2
x
,
0 <
x
< 1,
zero elsewhere.
Find the mean and variance of the sum
Y
of these four
random variables.
b)
4.1.8
Determine the mean and variance of the mean
X
of a random sample of size
9
from
a distribution having pdf
f
(
x
)
= 4
x
3
,
0 <
x
< 1,
zero elsewhere.
2.
a)
4.1.9
b)
4.1.13
a)
4.1.9
Let
X
and
Y
be random variables with
μ
1
= 1,
μ
2
= 4,
σ
1
2
= 4,
σ
2
2
= 6,
ρ
=
2
1
.
Find the mean and variance of
Z = 3
X – 2
Y.
b)
4.1.13
Determine the correlation coefficient of the random variables
X
and
Y
if
Var
(
X
)
= 4,
Var
(
Y
)
= 2,
and
Var
(
X + 2
Y
)
=
15.
3.
4.1.22
+
(c)
Let
X
be
N
(
μ
,
σ
2
)
and consider the transformation
X =
ln
(
Y
)
or, equivalently,
Y =
e
X
.
(a)
Find the mean and the variance of
Y
by first determining
E
(
e
X
)
and
E
[
(
e
X
)
2
]
,
by using the mgf of
X.
(b)
Find the pdf of
Y.
This is the pdf of the
lognormal distribution
.
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 Spring '08
 STEPANOV
 Statistics, Normal Distribution, Probability theory, maximum likelihood estimator, σ, θ

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