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EE{564,
Fall 2009
Homework 1
Due Monday, September 10, 2001
These problems have two main objectives: to recall key skills and tools from probability theory
and random processes and to develop a feel for comparing experimental/computer results with
calculated quantities.
1.
This problem addresses a method for converting two independent uniform random variables
into two independent Gaussian random variables. Consider the independent uniformly dis
tributed random variables
X
1
and
X
2
:
p
X
1
(
x
) =
p
X
2
(
x
) =
8
>
<
>
:
1
x
2
(0
;
1)
0
else
The purpose of this problem is to demonstrate that the following are independent Gaussian
random variables:
Y
1
=
q
2
ln
(
X
1
) cos(2
2
)
Y
2
=
q
2
ln
(
X
1
) sin(2
2
)
(a)
Determine the following,
p
X
1
;X
2
(
x
1
;x
2
)
,
E
f
Y
1
g
,
E
f
Y
2
g
, and
E
f
Y
1
Y
2
g
.
(b)
Consider the random variable,
R
=
q
2
ln
(
X
1
)
. Determine the pdf
p
R
(
r
)
of this
random variable and its mean.
(c)
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