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Final Exam: Probability
and Statistics (code 400178)
Vrije Universiteit Amsterdam
You are allowed to use a calculator, but not the book or your notes. You can express your numerical answers as fractions. Each question is worth 2 points (if answered completely and correctly). The total number of points is 40 (corresponding to 20 questions, divided in 8 problems). Give clear answers to as many questions as you can.
Problem 1. You and 9 other people are at small but fancy cocktail party where they serve 100 different cocktails. Suppose that each person is drinking a cocktail and that the different cocktails are all equally likely to be chosen.
a) What is the probability that two persons are drinking the same cock- tail?
b) What is the probability that someone else is drinking the same cocktail as you?
Problem 2. In a game of chance, your probability of winning is p. Suppose that you play three times, and consider the following events: A = {you win the first game}, B = {you win the second game} and C = {you win exactly once}.
a) For what value of p is C independent of A and B? Motivate your answer.
b) For 0 < p < 1, can the three events ever be mutually independent (i.e., P (A, B, C) = P (A)P (B)P (C))? Motivate your answer.
Problem 3. A die is thrown six times and the face values that come up are recorded. Let A = {the first value is 6}, B = {the second value is 6} and C = {6 comes up exactly once}.
a) Indicate all pairs of events that are independent? Motivate your answer.
b) Are the three events mutually independent? Motivate your answer.
Problem 4. A fair coin is flipped n times. You can give your answers as sums.
a) What is the probability to get exactly k heads?
b) What is the probability to get at least k heads?
c) Suppose that n is odd. What is the probability to get one more head than tails?
Problem 5. Let X be an exponential random variable with density function f (x) = λe−λx^ if x ≥ 0 and f (x) = 0 if x < 0.
a) Write the cumulative distribution function (cdf) F (x).
b) For what number y is P (X < y) = 1/2?
Problem 6. A box contains N identical balls. You extract k balls, mark them, and put them back inside the box. After shaking the box you extract k + 1 balls randomly, k of which are marked.
a) What is the probability of the event described?
b) Suppose that k = 1. What is your “maximum likelihood” estimate for N? Motivate your answer.
Problem 7. Suppose that the temperatures measured in Celsius at a par- ticular time of the year and geographic location are distributed according to a Normal distribution with μ = 30 and σ = 2. An American tourist planning to visit the place translates the temperatures into Fahrenheit by multiplying by 9/5 and then adding 32.
a) According to what type of distribution are the Fahrenheit temperatures distributed?
