Spring 2017, MATH 407, Mid-Term Exam 2
Monday, April 17, 2017
Instructor S. Lototsky (KAP 248D; x0–2389;
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Circle the time of your discussion section: 10am 11am
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 350. Compute approxi-
mately the probability that at most 100 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) or P (Z > r) (whichever applies), 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 three
customers arrive during the first hour. Compute the probability that at least one arrived during
the first 20 minutes. [To help you out: the arrival times are order statistics U(1), U(2), U(3) of
the uniform distribution on [0, 1], with time measured in hours. Your objective is to compute
the probability that the smallest of the three is less than 1/3, so it makes sense to go with the
complement: compute the probability that the smallest is bigger than 1/3.]
Problem 3. Let X, Y be independent random variables, both exponential with mean 1.
(a) Find the joint density of U = X + Y and V = X/(X + Y ).
(b) Are the random variables U and V independent? Explain your conclusion.
Problem 4. For a randomly selected group of 50 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
fX,Y (x, y) =
Cx, if x2 + y2 ≤ 1, x > 0, y ≥ 0,
0, otherwise.
(a) Are X and Y independent? Justify your answer.
(b) Compute E(X|Y ). Note: there is no need to know C.
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