STAT 410
Spring 2013
Homework #5
(due Friday, March 1, by 3:00 p.m.)
1.
Let the joint probability density function for
(
X
, Y
) be
f
(
x
,
y
)
=
3
y
x
,
0 <
x
< 2,
0 <
y
< 1,
zero
otherwise.
Let
U = X + Y
and
V = X
–
Y + 1.
Find the joint probability density function of
(
U, V
),
f
U,
V
(
u
,
v
).
Sketch the support of
(
U, V
).
2.
Let
X
and
Y
have the joint probability density function
f
X
,
Y
(
x
,
y
)
=
x
1
,
x
> 1,
0 <
y
<
x
1
,
zero
elsewhere.
Let
U = Y
and
V = Y
/
X.
Find the joint probability density function of
(
U, V
),
f
U,
V
(
u
,
v
).
Sketch the support of
(
U, V
).
3.
Let
> 0.
Consider two continuous random variables
X
and
Y
with joint p.d.f.
f
X,
Y
(
x
,
y
)
=
x
e
y
2
θ
2
θ
,
x
> 0,
0 <
y
<
x
2
.
Let
U = X
and
V =
X
/
Y
.
Find the joint probability density function of
(
U, V
),
f
U,
V
(
u
,
v
).
Sketch the support of
(
U, V
).
4.
2.7.4
Let
X
1
,
X
2
,
X
3
be
iid
with common
pdf
f
(
x
) =
e
–
x
,
x
> 0,
zero elsewhere.
Find the joint
pdf
of
Y
1
= X
1
,
Y
2
= X
1
+ X
2
,
Y
3
= X
1
+ X
2
+ X
3
.
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A company provides earthquake insurance.
The premium
X
is modeled by the
p.d.f.
f
X
(
x
)
=
5
2
5
x
e
x
,
0 <
x
<
∞
,
while the claims
Y
have the
p.d.f.
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 Spring '08
 Monrad
 Statistics, Normal Distribution, Probability, Probability theory, probability density function, $0.15, $0.09, $3.27

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