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University of Illinois
Fall 2010
ECE 313:
Problem Set 12: Solutions
Joint pdfs, Covariance, LLN and CLT
1.
[Joint pdfs of functions of random variables]
(a) The joint pdf of
X
and
Y
is given by:
f
X,Y
(
u,v
) =
(
1
(1

2
Q
(3))
√
2
π
(0
.
01)
e

(
v

0
.
3)
2
0
.
02
1
(0
.
4)
0
.
8
V
≤
u
≤
1
.
2
V,
0
≤
v
≤
0
.
6
V
0
else
where 1

2
Q
(3) is the probability a
N
(0
.
3
,
0
.
01) random variable falls in the interval [0
.
0
.
6]
.
Since
Q
(3) = 0
.
0013
,
the factor
1
1

2
Q
(3)
is approximately 1.0026, which is so close to one that the
distribution of
Y
is nearly Gaussian.
(b) Given:
α
=
k
1
u
(
u

v
)
β
=
k
2
u
u

v
We ﬁrst solve for
v
in terms of
u
from the second equation above to obtain
v
=
±
1

k
2
β
²
u
Then, we substitute for
v
in the ﬁrst equation to obtain
α
=
k
1
u
2

k
1
u
±
1

k
2
β
²
u
=
k
1
k
2
β
u
2
Keeping in the mind that
V
dd
and
V
t
are nonnegative, we get
u
=
r
αβ
k
1
k
2
v
=
±
1

k
2
β
²
r
αβ
k
1
k
2
Thus, (
W,Z
) =
g
(
X,Y
) is an invertible mapping because it is possible to obtain unique values of
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 Fall '08
 Milenkovic,O

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