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EE 351K Probability, Statistics, and Random Processes
SPRING 2009
Instructor: Shakkottai/Vishwanath
{
shakkott,sriram
}
@ece.utexas.edu
Homework 4 (Due 03/02/2009)
Problem 1
The runnerup in a road race is given a reward that depends on the difference between his
time and the winner’s time. He is given 50 dollars for being zero to two minutes behind, 30 dollars for being
two to ﬁve minutes behind, 10 dollars for being 5 to 10 minutes behind, and nothing otherwise. Given that
the difference between his time and the winner’s time is uniformly distributed between 0 and 15 minutes,
ﬁnd the mean and variance of the reward of the runnerup.
Problem 2
Let
X
be a random variable with PDF
f
X
(
x
) =
±
3
x
2
7
if
1
< x
≤
2
0
otherwise.
and let
Y
=
X
2
. Calculate
E
[
Y
]
and
var
(
Y
)
.
Problem 3
Let
X
be Gaussian with mean 3 and variance 5. Let
Y
= 2
X
+ 4
.
(a) Calculate the PDF of
Y
.
(b) Find
P
(
Y
≥
0)
.
Problem 4
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