STAT 410
Fall 2008
Homework #4
(due Friday, September 26, by 3:00 p.m.)
1.
Let X and Y have the joint probability density function
f
X,
Y
(
x
,
y
)
=
°
±
°
²
³
<
<
<
+
otherwise
0
1
0
4
x
y
y
x
a)
Find
f
Y
(
y
)
.
b)
Find
f
Y

X
(
y

x
)
.
c)
Find E
(
Y

X
)
.
d)
Are
X
and
Y
independent?
Justify your answer.
e)
Find Cov
(
X, Y
)
.
From the textbook:
2.3.3
Let
f
(
x
1
,
x
2
)
=
3
2
2
1
21
x
x
, 0 <
x
1
<
x
2
< 1, zero elsewhere, be the joint
pdf
of X
1
and X
2
.
(a) Find the conditional mean and variance of X
1
, given X
2
=
x
2
, 0 <
x
2
< 1.
(b) Find the distribution of Y = E
(
X
1

X
2
)
.
(c) Determine E
(
Y
)
and Var
(
Y
)
and compare these to E
(
X
1
)
and Var
(
X
1
)
,
respectively.
2.3.10
Let X
1
and X
2
have joint
pmf
p
(
x
1
,
x
2
)
described as follows:
(
x
1
,
x
2
)
(
0,
0
)
(
0,
1
)
(
1,
0
)
(
1,
1
)
(
2,
0
)
(
2,
1
)
p
(
x
1
,
x
2
)
18
1
18
3
18
4
18
3
18
6
18
1
and
p
(
x
1
,
x
2
)
is equal to zero elsewhere. Find the two marginal probability
mass
functions and the two conditional means.
Hint:
Write the probabilities in a rectangular array.
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2.3.11
Let us choose at random a point from interval
(
0,
1
)
and let the random variable X
1
be equal to the number which corresponds to that point. Then choose a point at random
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 Fall '08
 AlexeiStepanov
 Probability, Probability theory, probability density function, joint PDF

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