EE131A: Practice Problems for Final Exam
The exam will have 6 questions
More than 6 problems have been provided for practice purposes
Problem 1.
(
25 pts
)
(a) (10 pts) Students taking a test have to answer 4 out of 5 questions, not in any specific order. The questions are
numbered 1–5. Assuming that a student chooses questions randomly, what is the probability that she will answer
questions 2–5 (again, not in any specific order)?
(b) (15 pts) Now suppose that the questions are multiple choice, and each question has 4 possible answers, out
of which
exactly one
choice is the correct answer. For each correct answer, students get 10 points, but for each
incorrect answer they lose 2 points. One of the students has not studied at all and he is just guessing his answers.
What is the probability that he will score at least 20 points on the 4 questions he answers?
Sol.
(a) P(problem 2,3,4,5 is choosen)=
1
0
@
5
4
1
A
=
1
5
(b) If the students gives 0 correct answer, he gets
(

2)(4) =

8
points; else if he gives 1 correct answer, he
gets
10 + (

2)(3) = 4
points; else if he gives 2 correct answer, he gets
20

4 = 16
points; else if he gives 3
correct answer, he gets
30

2 = 28
points.
..
So, P(he scores at least 20 points) = P(he gives at least 3 correct answer) =
p
4
3
P
p
3
(1

p
) +
p
4
=
4(0
.
25
3
)(0
.
75) + 0
.
25
4
= 0
.
05
Problem 2.
(25 pts)
(a) (15 pts) A continuous random variable
X
that takes on only positive values (i.e.,
P
[
X
≤
0] = 0
) has cdf
F
X
(
x
)
and pdf
f
X
(
x
)
. Find the cdf and pdf of the random variable
Y
= ln
X
in terms of
F
X
(
x
)
and
f
X
(
x
)
.
(b) (10 pts) In a large hotel it is known that each guest has a
1%
chance of losing his key. If a key is lost,
the guest must pay a fine of $20. A group of 200 UCLA students checks into the hotel for a meeting.
Using the
Poisson approximation
, compute the probability that the hotel will collect at least $60 in lost key fines from the
group.
Sol.
(a)
F
Y
(
y
) =
P
(
Y
≤
y
) =
P
(ln
X
≤
y
) =
P
(
X
≤
e
y
) =
F
X
(
e
y
)
f
Y
(
y
) =
dF
Y
(
y
)
dy
=
F
p
X
(
e
y
)
d
dy
e
y
=
f
X
(
e
y
)
e
y
(b) Let X be the the number lost keys by all the 200 students. Then
α
=
E
[
X
] = 200
×
0
.
01 = 2
. For the
hotel to collect at least $60, at least three students must lose their keys, or
X
≥
3
. Hence,
P
(
X
≥
3) =
e

2
∞
s
k
=3
2
k
k
!
= 1

e

2
2
s
k
=0
2
k
k
!
= 1

5
×
e

2
= 0
.
32332
Note that if no approximation is made, then we have to use Binomial law, and the calculations will go as
follows:
P
(
at least $60 in fines
) =
P
(
X
≥
3) = 1

2
s
i
=0
p
200
i
P
(0
.
01)
i
(0
.
99)
200

i
,
1