STAT 410
Fall 2011
Homework #4
(due Friday, September 23, by 3:00 p.m.)
1.
Let
X
be a
Uniform
(
0,
1
)
and
Y
be a
Uniform
(
0,
3
)
independent
random
variables.
Let
W = X + Y.
Find and sketch
the
p.d.f.
of
W.
2.
Let random variables
X
and
Y
have
an Exponential distribution with mean 1
and
a Uniform distribution on
(
0
,
1
)
,
respectively.
Suppose
X
and
Y
are
independent.
Find the probability density function of
W = X + Y.
3.
Let
X
and
Y
have the pdf
f
(
x
,
y
)
= 1,
0 <
x
< 1,
0 <
y
< 1,
zero elsewhere.
Find the
c.d.f.
and the
p.d.f.
of
W
=
X – Y.
“Hint”:
Consider two cases:
–
1 <
w
< 0
and
0 <
w
< 1.
4.
Let
X
and
Y
have the pdf
f
(
x
,
y
)
= 1,
0 <
x
< 1,
0 <
y
< 1,
zero elsewhere.
Find the
c.d.f.
and the
p.d.f.
of
V
=
X
/
Y
.
“Hint”:
Consider two cases:
0 <
v
< 1
and
v
> 1.
5.
Let X and Y have the joint p.d.f.
f
X
Y
(
x
,
y
)
=
20
x
2
y
3
,
0 <
x
< 1,
0 <
y
<
x
,
zero elsewhere.
Let
U = Y
2
and
V = X
Y.
Find the joint probability density function of
(
U, V
)
,
f
U
V
(
u
,
v
)
.
Sketch the support of
(
U, V
)
.
6.
Suppose the joint probability density function of
(
X
, Y
)
is
f
X
Y
(
x
,
y
)
=
≤
≤
≤
otherwise
0
1
0
15
2
x
y
y
x
Let
U = X + Y
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 Fall '08
 Monrad
 Statistics, Probability, Probability theory, probability density function

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