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EE 351K Probability, Statistics, and Random Processes
SPRING 2009
Instructor: Shakkottai/Vishwanath
{
shakkott,sriram
}
@ece.utexas.edu
Homework 5 (Due 03/09/2009)
Problem 1
Consider two continuous random variables
Y
and
Z
and a random variable
X
that is equal
to
Y
with probability
p
and to
Z
with prbability
1

p
. Show that the PDF of
X
is given by
f
X
(
x
) =
pf
Y
(
x
) + (1

p
)
f
Z
(
x
)
Problem 2
The random variable
X
has the PDF
f
X
(
x
) =
±
cx

2
,
1
≤
x
≤
4
0
,
otherwise
1. Determine the value of
c
.
2. Let
A
be the event
{
X >
2
}
. Calculate
P
(
A
)
and the conditional PDF of
X
given that event
A
has occured.
Problem 3
Let
X
and
Y
be two identical, but independent exponential random variables with PDFs
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Unformatted text preview: λeλx and λeλy respectively. Let Z = X + Y . Find the conditonal PDF of X given Z = z . Problem 4 Let X and Y be two random variables that are independent and uniformly distributed over the interval [0 , 1] . Find the CDF and PDF of  XY  . Problem 5 Find c and the PDF of the continuous random variable X associated with the moment generating function M X ( s ) = c. 3 3s + 3 4 . 7 7s ....
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This note was uploaded on 10/26/2009 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas at Austin.
 Spring '07
 BARD

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