EE 351K Probability, Statistics, and Random Processes
SPRING 2009
Instructor: Shakkottai/Vishwanath
shakkott, [email protected]
Homework 4  Solutions
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.
Solution :
Let
X
be the reward. The probability that
X
= 50
is
2
15
, the probability that
X
= 30
is
3
15
, and the
probability that
X
= 10
is
5
15
. Therefore
E
[
X
] =
2
15
·
50 +
3
15
·
30 +
5
15
·
10 = 16
,
and
var
(
X
) =
E
[
X
2
]

(
E
[
X
])
2
=
2
15
·
50
2
+
3
15
·
30
2
+
5
15
·
10
2

(16)
2
= 290
.
6667
.
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
)
.
Solution:
We have
E
[
Y
] =
E
[
X
2
] =
Z
2
1
x
2
f
X
(
x
)
dx
=
Z
2
1
3
x
4
7
dx
=
3
5
x
5
7
²
²
²
2
1
=
3
5
(
32
7

1
7
) =
93
35
= 2
.
6571
To obtain the variance of
Y
, we ﬁrst calculate
E
[
Y
2
]
. We have, after straightforward calculation,
E
[
Y
2
] =
E
[
X
4
] =
Z
2
1
x
4
f
X
(
x
)
dx
=
Z
2
1
3
x
6
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 Spring '07
 BARD
 Probability theory, Random Processes Instructor

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