Fall 2021, MATH 407, Mid-Term Exam 2
Wednesday, November 17, 2021, 9:00-9:50am
Instructor S. Lototsky (KAP 248D; x0–2389;
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Instructions:
• No books, notes, calculators, or help from other people.
• Turn off cell phones.
• Show your work/explain your answers.
• You have 50 minutes to complete the exam.
• There are five problems; 10 points per problem.
• Upload the solutions to GradeScope.
standard normal pdf: (2π)−1/2e−x
2/2; Gamma(a, b) pdf: ba
xa−1e−bx; Exponential with
mean θ is Gamma(1, 1/θ), Beta(a, b) pdf:
xa−1(1−x)b−1; Poisson, mean µ, pmf: e−µkµ/k!.
Problem 1. For a randomly selected group of 50 people, compute the expected number of dis-
tinct birthdays (that is, the expected number of the days of the year that are a birthday of at least
one person in the group). Assume 365 days in a year.
Problem 2. 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.
Compute E(X|Y ). Note: there is no need to know C.
Problem 3. At a particular location, there is, on average, one earthquake every 4 days. Assuming
that the earthquakes follow Poisson process, compute, approximately, the probability that there are
more than 100 earthquakes in 360 days. Leave your answer in the form P(N < r) or P(N > r),
where N is a standard normal random variable and r is a real number. Then circle the interval
that contains your answer:
(0, 0.1) [0.1, 0.3) [0.3, 0.5) [0.5, 1)
Problem 4. Let X,Y be independent exponential random variables with E(X) = E(Y ) = 1/2.
Compute the probability density functions of the random variables X + Y and X/(X + Y ).
Problem 5. Customers arrive at a bank at a Poisson rate λ. Suppose that two customers arrive
during the first hour. Compute the probability that at least one of the customers arrived during
the first 15 minutes.
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