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Final Exam: Probability
and Statistics (code 400178)
Vrije Universiteit Amsterdam
You are allowed to use a calculator, but the calculations involved in solving the problems can all be carried out without one. You can express your numerical answers as fractions. In carrying out the calculations of Problems 4 and 5, you can for simplicity replace the number 1/e with 1/3. Each question is worth 2 points (if answered completely and correctly). The total number of points is 40 (corresponding to 20 questions, divided in 6 problems). Give clear answers to as many questions as you can.
Problem 1. In conducting an experiment, a temperature is measured several times and it is found that the values are distributed according to a Normal distribution with μ = 15 Celsius and σ = 0.1 Celsius. To convert a Celsius temperature to Fahrenheit, you have to multiply by 9/5 and then add 32.
a) What is the distribution of the temperature expressed in Fahrenheit?
b) What are the expected value and the variance of the temperature ex- pressed in Fahrenheit?
Problem 2. Let X be a uniform random variable on [a, b].
a) What is the density function f (x) of X?
b) Calculate the expected value and the variance of X.
c) Find x 0 such that P (X > x 0 ) = 1/3.
d) A line segment of length 1 is cut once at random. What is the proba- bility that the longer piece is more than twice the length of the shorter piece?
Problem 3. Consider a dice with the faces 2, 3, 5 colored red and the other faces colored green.
a) What is the expected number of times you have to roll the dice to get a red face with an even number on it?
b) If the dice is rolled n times, what is the probability of obtaining at least m times a red face with an even number on it? (m ≤ n)
c) The dice is rolled twice. Are the events A = {two even numbers come out} and B = {red comes out at least once} independent? Compute P (A|B).
d) The dice is rolled three times. What is the probability of getting red two or more times given that red came out at least once?
e) The dice is rolled three times. What is the probability of getting red two or more times given that three even numbers came out?
f) Each of 3 persons rolls the dice once. What is the probability that two or more persons obtain the same number?
Problem 4. Assume that the waiting time of a job in a printer’s queue can be modelled by an exponential random variable with density
f (t) =
λe−λt, t ≥ 0 0 , t < 0
where time is measured in minutes.
a) What is the probability that a job has to wait for longer than a time τ = 1/λ?
b) Suppose that you send a job to the printer at 10:30. At 10:45 your job has not been printed yet. How much longer do you have to wait (approximately) to have a probability of approximately two thirds that your job has been printed? Express your answer in terms of λ.
c) At 11:00 your job has not been printed yet. Answer question b) again. Express your answer in terms of λ.
