Fall 2013, MATH 407, Final Exam
December 16, 2013
Instructor — S. Lototsky (KAP 248D; x0–2389;
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Instructions:
• No books, notes, or calculators.
• You have 120 minutes to complete the exam.
• Show your work.
Problem Possible Actual Problem Possible Actual
5 20 10 20
Total 100 Total 100
Problem 1. A total of 10 identical gifts are distributed at random among 7 children. Compute
the probability that every child receives at least one gift.
Problem 2. Suppose that A and B are independent events for which P (A) = 0.3 and P (B) = 0.4.
What is the probability that either A or B occurs, but not both?
Problem 3. A population contains twice as many females as males. In this population, 5% of
males and 0.25% of females are color-blind. A color-blind person is selected at random. Compute
the probability that the person is male.
Problem 4. Compute the proportion of all the four-children families with more girls than boys.
Assume that that boys and girls are equally likely.
Problem 5. Let X be a standard normal random variable. Define the random variable Y by
Y = eX . Compute the probability density function of the random variable Y .
Problem 6. For a randomly selected group of 100 people, denote by X the number of days in
a 365-day year that are not a birthday of any person in the group. Compute the expected value of
Problem 7. Let X and Y be independent standard random variables. Explain why the random
variables X + Y and X − Y are independent.
Problem 8. Let X, Y be independent random variables, both exponentially distributed with
(a) Compute the joint density of U = X + Y and V = X/(X + Y ).
(b) Are random variables U and V independent? Explain your answer.
Problem 9. A fair die is rolled until the total sum of all rolls exceeds 290. Compute approxi-
mately the probability that at most 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 10. Customers arrive at a bank according to a Poisson process. Suppose that three
customers arrived during the first hour. Compute the probability that nobody arrived during the
first 15 minutes.
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