STAT 410
Spring 2008
Homework #3
(due Friday, February 8, by 3:00 p.m.)
1.
Suppose that the random variables X and Y have joint
p.d.f.
f
(
x
,
y
) given by
f
(
x
,
y
) =
C
x
2
y
,
0 <
x
<
y
,
x
+
y
< 2.
a)
Sketch the support of
(
X
, Y
).
b)
What must the value of
C
be so that
f
(
x
,
y
) is a valid joint
p.d.f.?
c)
Find P
(
X + Y < 1
).
2.
Suppose that the random variables X and Y have joint
p.d.f.
f
(
x
,
y
) given by
f
(
x
,
y
) = 6
x
2
y
,
0 <
x
<
y
,
x
+
y
< 2.
a)
Find the marginal probability density function for X.
b)
Find the marginal probability density function for Y.
From the textbook:
2.1.6
Let
f
(
x
,
y
) =
e
–
x
–
y
, 0 <
x
<
∞
, 0 <
y
<
∞
, zero elsewhere, be the pdf of
X
and
Y
.
Then if
Z
=
X
+
Y
, compute
P
(
Z
≤
0
),
P
(
Z
≤
6
), and, more generally,
P
(
Z
≤
z
), for
0 <
z
<
∞
. What is the pdf of
Z
?
2.1.7
Let
X
and
Y
have the pdf
f
(
x
,
y
) = 1, 0 <
x
< 1, 0 <
y
< 1, zero elsewhere. Find the
cdf and pdf of the product
Z
=
X
Y
.