SPRING 2009
MATH 425
Ralf Spatzier
Homework 5
for Wednesday, June 10
(1) The random variables
X
and
Y
have the joint density function
f
(
x,y
) = 12
x y
(1

x
) for )
< x <
1
,
0
< y <
1
and equal to 0 otherwise.
(a) Are
X
and
Y
independent?
(b) Find
E
[
X
].
(c) Find
E
[
Y
].
(d) Find
V ar
(
X
).
(e) Find
V ar
(
X
).
(2) Three points
X
1
,X
2
,X
3
are selected a t random on a line
L
of length 1. What is the
probability that
X
2
lies between
X
1
and
X
3
?
(3) Let
X
and
Y
have joint density function
f
(
x,y
) =
1
x
2
y
2
x
≥
1
,y
≥
1
.
(a) Compute the joint density function of
U
=
XY
and
V
=
X/Y
.
(b) What are the marginal densities of
U
and
V
?
(4) The gross daily sales at Amer’s is a normal random variable with mean $2
,
200 and
standard deviation $230. What is the probability that
(a) that the total gross sales over the next 2 days exceed $5,000?
(b) That the daily sales exceed $2,000 in at least 2 of the next 3 days?
(5) Suppose
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 Spring '10
 k
 Math, Probability theory, joint density function, Ralf Spatzier, radie 1,2

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