Fall 2021, MATH 407, Final Exam
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 120 minutes to complete the exam.
• There are 10 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. A box contains 12 blue balls, 17 green balls and 21 red balls, [50 total, well mixed].
15 balls are taken out of the box, all at once.
Compute the probability that 6 of those balls are blue, 5 are green, and 4 are red. Do not
simplify/evaluate binomial coefficients.
(12 choose 6)(17 choose 5)(21 choose 4)/(50 choose 15)
Problem 2. In a certain town, there are twice as many cars as trucks, and 3% of cars and 1%
of trucks are of yellow color. A vehicle is selected at random, and has yellow color. Compute the
probability that the vehicle is a car.
0.03 ∗ (2/3)/(0.03 ∗ (2/3) + 0.01 ∗ (1/3)) = 6/7
Problem 3. A charitable lottery has 20,000 tickets, of which 400 win prizes and the rest win
nothing. You buy 50 tickets.
(a) Compute the number of the prize-winning tickets you expect to find.
(b) Which of the following numbers is closest to the probability that, out of 50 tickets, at least
one is prize-winning:
Explain your conclusion.
Poisson(1) > 0: 1-(1/e) = 0.632, closest to 2/3.
Problem 4. Let X be a random variable with uniform distribution on [−2, 2]. Define the random
variable Y by Y = ln(2−X). Compute the probability density function of the random variable Y .
Make sure to indicate the range of possible values for Y .
P (Y < x) = P (X > 2− ex) = ex/4, x < ln 4; fY (x) = ex/4I(x < ln 4)
Problem 5. For a randomly selected group of 60 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.
365 * P(somebody is born on Jan 1)
= 365 (1- P(nobody born Jan 1)) = 365 (1- (364/365)60)
Problem 6. The joint probability density function of two random variables X and Y is
fX,Y (x, y) =
Cxy, if x2 + y2 ≤ 1, x ≥ 0, y ≥ 0,
0, otherwise.
Compute the conditional expectation E(X|Y ). Note: there is no need to know C.
pdf of Y is (Cy/2)(1− y2);E(X|Y ) = (2/3)(1− Y 2)1/2
Problem 7. 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
at least 90 earthquakes in 320 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)
Poisson(80) > 89.5 [use continuity correction] or Gamma(90, 1/4) < 320;
either way, approximately P (N > 1.055) ∈ [0.1, 0.3)
Remember that the variance of Poisson random variable is equal to the mean (in this
case, 80),
and Gamma(90, 1/4) is the sum of 90 iid exponential random variables, each having
the mean and standard deviation equal to 4 (days)
Problem 8. Let X,Y be independent exponential random variables, both with mean value
equal to 1/3. Compute the probability density functions of the random variables U = X + Y and
V = X/(X + Y ). Do not forget to indicate the range of the random variables U
pdfs of X and Y are 3e−3x, so fU(x) = 9xe
−3x, x > 0; V is uniform on (0, 1)
Problem 9. Customers arrive at a bank at a Poisson rate λ. Suppose that five customers arrive
during the first hour. Compute the probability that at least one of the customers arrived during
the last 15 minutes. The final answer should be in the form a/b for some positive
numbers a < b.
1− P (all five are in the first 3/4 of the hour) = 1− (3/4)5 = 781/1024
Problem 10. Let X1, X2, . . . be iid standard normal random variables.
(1) Identify the distribution of n−1/2
k=1Xk. Normal mean 0, variance 1, i.e. also
standard normal
(2) Compute the limit
= EX21 = 1
(3) By completing the square, compute the limit
Here we use the results of (1) and (2), together with the Slutsky theorem:
convergence in (2) is with probability one. Recall that χ21 is the square of
the standard normal random variable.
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