Final Exam
This is an old final exam from a similar course I taught at Duke.
Note that it was a 3 hour exam, so quite a bit longer than ours, and
that problem 7 is from section 6.5, which we did not cover.
• Do not o
...
Final Exam
This is an old final exam from a similar course I taught at Duke.
Note that it was a 3 hour exam, so quite a bit longer than ours, and
that problem 7 is from section 6.5, which we did not cover.
• Do not open this test booklet until you are directed to do so.
• You will have 3 hours to complete the exam.
• This is a closed book exam. No calculators, notes, or any other form of assistance is allowed.
Read each question carefully and answer it in the space provided. YOU MUST SHOW
ALL OF YOUR WORK TO RECEIVE FULL CREDIT! Please be clear and concise in your
solutions.
• In problems involving the standard normal cumulative distribution function Φ, you may leave
your answer in the form Φ
√a−bc
def
; that is, you don’t have to simplify the fraction or evaluate
what is under a square root sign.
Problem Points Score
1 20
2 18
3 15
4 15
5 10
6 12
7 10
Total 100
Name:
1
Math 135
May 1, 2012
Final Exam
Question 1 (20 pts). Suppose (X, Y ) have joint density
f(x, y) =
ce−2y
if 0 < x < y
0 otherwise.
a) Find c.
b) Find E[XeY
].
c) Compute the marginal density fX(x).
d) Compute P(X + Y < 2)
2
Math 135
May 1, 2012
Final Exam
Question 2 (18 pts). A niche video store sells two types of movies: zombie movies and vampire
movies. People tend to strongly prefer one genre over the other, so we can classify the customers
into zombie-lovers and vampire-lovers. Assume that these two types of customers arrive at the store
according to two independent Poisson processes. Since vampires are more popular lately, vampire
customers arrive at a rate of 3 per hour, while zombie customers arrive at a rate of only 2 per hour.
a) Find the probability that exact
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