Fall 2013, MATH 407, Mid-Term Exam 2
Wednesday, November 20, 2013
Instructor S. Lototsky (KAP 248D; x0–2389;
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Circle the time of your discussion section: 8am 9am 10am
Instructions:
• No books, notes, or calculators.
• You have 50 minutes to complete the exam.
• Show your work.
Problem Possible Actual
Problem 1. A fair die is rolled until the total sum of all rolls exceeds 300. Compute approx-
imately the probability that at least 80 rolls are necessary. Note that, for a single roll of the die,
the expected value and variance of the outcome are 7/2 and 35/12, respectively. Use the continuity
correction. Leave the answer in the form P (Z < r), where Z is a standard normal random variable
and r is a suitable real number.
Problem 2. Customers arrive at a bank according to a Poisson process. Suppose that two
customers arrives during the first hour. Compute the probability that at least one arrived during
the first 20 minutes.
Problem 3. Let X, Y be independent random variables, both uniform on (0, 1). Find the joint
density of X + Y and X/(X + Y ).
Problem 4. For a randomly selected group of 100 people, compute the expected number of
distinct birthdays (that is, the expected number of the days of the year that are a birthday of at
least one person in the group).
Problem 5. The joint probability density function of two random variables X and Y
fXY (x, y) =
Cy, if x2 + y2 ≤ 1, |x| ≤ 1, y ≥ 0,
0 otherwise.
(a) Are X and Y independent? Justify your answer.
(b) Compute E(X|Y ). Suggestion: keep your computations to a minimum. In particular, there
is no need to know C.
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