EE 351K Probability and Random Processes
FALL 2010
Instructor: Prof. Haris Vikalo
[email protected]
Homework 4 Solutions
Problem 1
There are
n
multiplechoice questions in an exam, each with 5 choices. The student knows
the correct answer to
k
of them, and for the remaining
n

k
guesses one of the 5 randomly. Let
C
be the
number of correct answers, and
W
be the number of wrong answers.
(a) What is the PMF of
W
? Is
W
one of the common random variables we have seen in class?
(b) What is the PMF of
C
? What is its mean,
E
[
C
]
?
Solution:
(a) Assuming the student does not intentionally answer any of the questions he knows incorrectly, we
have that
W
indicates the number of “successes” of a Benoulli random variable, so that
P
(
W
=
l
) =
n

k
l
4
5
l
1
5
n

k

l
.
We have, then, that
W
is distributed as a Binomial random variable with parameters
(
n

k,
4
/
5)
.
(b) We have that
C
=
n

W
⇒
W
=
n

C
. It follows, then, that
P
(
C
=
l
) =
P
(
W
=
n

l
)
=
n

k
n

l
4
5
n

l
1
5
n

k

(
n

l
)
=
n

k
n

l
4
5
n

l
1
5
l

k
.
Now,
E
[
C
] =
E
[
n

W
] =
n

E
[
W
]
.
Since
W
is a binomial random variable, we have that
E
[
W
] = (
n

k
)
4
5
. Then
E
[
C
] =
n

4
5
(
n

k
) =
1
5
(
n
+ 4
k
)
.
Problem 2
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 20 dollars for being one minute behind, 10 dollars for being one
to two minutes behind, 5 dollars for being 2 to 6 minutes behind, and nothing otherwise. Given that the
difference between his time and the winner’s time is uniformly distributed between 0 and 12 minutes, find
the mean and variance of the reward of the runnerup.
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