IE 300 / GE 331 Spring 2009
Homework #5
Due: April 9th
Problem 1.
Random variables
X
and
Y
have the joint PDF shown below:
(a) Prepare neat, fully labeled sketches of
f
X

Y
(
u

v
).
(b) Find
E
[
X

Y
=
v
] and var[
X

Y
=
v
].
(c) Find
E
[
X
].
Problem 2.
X
and
Y
denote
independent
standard Gaussian random variables.
(a) What is the joint pdf
f
X
,
Y
(
u,v
) of
X
and
Y
?
1
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2
(b) Sketch the
u

v
plane and indicate on it the region over which you need to
integrate the joint pdf in order to ﬁnd
P
{X
2
+
Y
2
>
2
α
2
}
. Compute
P
{X
2
+
Y
2
>
2
α
2
}
.
(c) Let
Z
=
X
2
+
Y
2
. What is the pdf of
Z
?
(d) Express
P
{X
> α
}
in terms of the complementary unit Gaussian CDF func
tion
Q
(
x
), and use this to write
P
{X
> α,
Y
> α
}
in terms of
Q
(
x
). (Re
member commas mean intersections).
(e) On your sketch of part (b), show the region over which you must integrate the
joint pdf to ﬁnd
P
{X
> α,
Y
> α,
}
. Use your sketch to prove the following
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 Spring '09
 NegarKayavash
 Normal Distribution, Variance, Cumulative distribution function, joint PDF

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