EE 351K PROBABILITY & RANDOM PROCESSES
FALL 2011
Instructor: Sujay Sanghavi
[email protected]
Homework 4 Solution
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
]
?
Sol
:
(a) The student guesses one of the 5 choices randomly, probability that a question is guessed correctly
=
1
5
.
Since the student knows
k
answers correctly, the number of wrong answers
W
∈
[0
, n

k
]
. Therefore
P
W
(
w
) =
{ (
n

k
w
)
(
4
5
)
w
(
1
5
)
n

k

w
w
∈
[0
, n

k
]
0
otherwise
which is a binomial random variable.
(b) Similarly the PMF of
C
is
P
C
(
c
) =
{ (
n

k
c

k
)
(
1
5
)
c

k
(
4
5
)
n

c
c
∈
[
k, n
]
0
otherwise
Then
E
[
C
] =
n
∑
c
=
k
c
(
n

k
c

k
)
(
1
5
)
c

k
(
4
5
)
n

c
=
k
+
n

k
5
.
Problem 2
Fischer and Spassky play a suddendeath chess match whereby the first player to win a game wins the
match. Each game is won by Fischer with probability
p
, by Spassky with probability
q
, and is a draw with
probability
1

p

q
.
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 Spring '07
 BARD
 Probability theory, Conditional expectation

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