Spring 2017, MATH 407, Final Exam
May 3, 2017
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 box contains 10 blue balls, 11 red balls, and 12 green balls [33 total, well mixed].
15 balls are taken out of the box, all at once. What is the probability that at least one of those 15
balls is red?
Problem 2. Imagine yourself making a salad by selecting one ingredient from each of the
following groups:
Group 1: Potato, Yam;
Group 2: Cucumber, Carrot, Reddish;
Group 3: Garlic, Onion, Parsly.
Within every group, each ingredient is equally likely to be selected, and the selections are inde-
pendent from group to group.
What is the probability that your salad will contain potato or parsley, but not both?
Problem 3. A population contains twice as many males as females (that is, two males for each
female). In this population, 5% of males and 0.5% of females are color-blind. A color-blind person
is selected at random. Compute the probability that the person is female.
Problem 4. Let X be a standard Gaussian random variable. Define the random variable Y by
X, if |X| < 1;
−X, if |X| ≥ 1.
Determine the probability density fY function of Y . [Use the definition of Y to conclude that
P (Y ≤ t) = P (X ≤ t) for all t. Keep in mind that P (X > −t) = P (X < t) for all t. Your final
answer should be a probability density function.]
Problem 5. Let X be a random variable with uniform distribution on [2, 5]. Define the random
variable Y by Y = − ln(X − 2). Compute the probability density function of the random variable
Problem 6. Let U be uniform random variable on (−π, π) and let V be exponential random
variable with mean 1. Assume that U and V are independent. Define the random variables X =√
2V cos(U), Y =
2V sin(U).
(a) Compute the joint pdf of X and Y .
(b) Are X − Y and X + Y independent? Explain your conclusion?
Problem 7. Four balls are dropped at random into five boxes so that the balls are dropped
independently of one another and each ball is equally likely to land in any of the boxes. Denote by
X the number of empty boxes. Compute the average number E(X) of empty boxes and make sure
that your answer is bigger than one.
Problem 8. At a certain location, there is, on average, one earthquake per day. Assume that
the earthquakes follow a Poisson process.
(a) Compute approximately the probability to have fewer than 66 earthquakes in 64 days. Use
the Central Limit Theorem; depending on the way you set up your approximation, you may need
to use continuity correction. Leave your answer in the form P (Z < a) or P (Z > b), where Z is a
standard normal random variable.
(b) Is the answer you got in part (a) bigger than 1/2 or less than 1/2? Explain your conclusion.
Problem 9. Customers arrive at a bank at a Poisson rate λ. Suppose that three customers
arrived during the first hour. Compute the probability that at least one arrived during the last 15
Problem 10. Customers arrive at a bank at a Poisson rate λ. Suppose that five customers
arrived between noon and 1pm. Let X be the arrival time of the last customer. Compute E(X),
the expected value of X. [If you have no idea what to do, compute E max(U1, U2, U3, U4, U5) where
U1, U2, U3, U4, U5 are iiid uniform on (0, 1). Start by computing P (max(U1, U2, U3, U4, U5) ≤ x). For
full credit, you final answer should be expressed as a time moment between noon and 1pm.]
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