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Student Number Queen’s University Department of Mathematics and Statistics STAT/MTHE 353 Final Examination April 21, 2012 Instructor: T. Linder • PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam questions as written.” • This material is copyrighted and is for the sole use of students registered in MTHE/STAT 353 and writing this examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright and may also constitute a breach of academic integrity under the University Senate’s Academic Integrity Policy Statement. • Formulas and tables are attached. • An 8.5 × 11 inch sheet of notes (both sides) is permitted. • Simple calculators (Casio 991, blue or gold sticker) are permitted. HOWEVER, do reasonable simplifications. • Write the answers in the space provided, continue on the backs of pages if needed. • SHOW YOUR WORK CLEARLY. Correct answers without clear work showing how you got there will not receive full marks. Marks: Please do not write in the space below. Problem 1 [10] Problem 4 [10] Problem 2 [10] Problem 5 [10] Problem 3 [10] Problem 6 [10] Total: [60] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 2 of 12 1. Let X1 , X2 , and X3 be independent exponentially distributed random variables with parameter 1. Define Y1 = X1 + X2 , Y2 = X1 + X3 , and Y3 = X2 + X3 . (a) Find the joint probability density function of Y1 , Y2 , Y3 . [6] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 3 of 12 (b) Find the probability P (Y1 > Y2 > Y3 ). Hint: Do not integrate the joint pdf of Y1 , Y2 , Y3 . [4] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 4 of 12 2. Suppose we are given a random N number of balls, where N is a Poisson random variable with parameter λ. We distribute these N balls randomly into n urns in such a way that each ball is equally likely to be placed in any of the n urns, independently of the placement of the other balls. Compute the expected number of empty urns. [10] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 5 of 12 3. Suppose that at a dinner party attended by n people, each of the n2 pairs of people are, independently, friends with probability p. Let Xi , i = 1, . . . , n denote the number of friends at the dinner party of the ith individual. (a) What is the distribution of Xi ? [2] (b) Find ρ(Xi , Xj ), the correlation coefficient between Xi and Xj for i 6= j. [8] STAT/MTHE 353 – Final Exam, April 21, 2012 4. (a) Let X and Y be random variables having finite variances. Show that Cov X, E[Y |X] = Cov(X, Y ). Page 6 of 12 [5] (b) Suppose that Var(X) > 0 and for some real constants a and b, E[Y |X] = a + bX. Cov(X, Y ) . [5] Show that b = Var(X) STAT/MTHE 353 – Final Exam, April 21, 2012 Page 7 of 12 5. (a) Suppose that the mean lifetime of a certain type of cheap light bulb is 100 hours with a standard deviation of 30 hours. You light a room with a lamp that uses this type of light bulb, and when a light bulb fails, you immediately replace it. Use the central limit theorem to estimate how many of these light bulbs you should store so that the probability that you can continually light the room for the next 2000 hours is at least 0.95. [6] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 8 of 12 (b) Let Y1 , Y2 , . . . be a sequence of independent Bernoulli(1/2) random variables and define Xn = 2n Y1 Y2 · · · Yn . Does Xn converge in probability, and if yes, to what limit? [4] Bonus: Does Xn converge almost surely, and if yes, to what limit? STAT/MTHE 353 – Final Exam, April 21, 2012 Page 9 of 12 6. Let Y1 , Y2 , . . . be independent and identically distributed discrete random variables taking values in {0, 1, . . . , 9} with probabilities P (Yi = k) = 1/10 for k = 0, 1, . . . , 9. Let X be a continuous random variable that is uniformly distributed on the interval [0, 1]. (a) Find the moment generating function MYi (t) of the Yi ’s and the moment generating function MX (t) of X. [4] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 10 of 12 P (b) Let Xn = ni=1 10−i Yi . Use the Lévy continuity theorem for moment generating functions to show that Xn converges to X in distribution as n → ∞. [6] STAT/MTHE 353 – Final Exam, April 21, 2012 Page 11 of 12 Formula Sheet Sums ∞ X xk = k=0 1 , if |x| < 1, 1−x N X xk = k=0 1 − xN +1 if x 6= 1. 1−x Special Distributions P (X = 0) = 1 − p, P (X = 1) = p E(X) = p, Var(X) = p(1 − p). Bernoulli(p): Binomial(n, p) : Geometric(p): n k P (X = k) = p (1 − p)n−k , k = 0, 1, 2, . . . , n, k E(X) = np, Var(X) = np(1 − p). P (X = k) = p(1 − p)k−1 , k = 1, 2, . . . . E(X) = 1/p, Var(X) = (1 − p)/p2 . λk −λ e , k = 0, 1, 2, . . . , k! E(X) = λ, Var(X) = λ. Poisson(λ): P (X = k) = Continuous uniform on [a, b]: 1/(b − a) if a ≤ x ≤ b f (x) = 0 otherwise, E(X) = a+b , 2 Var(X) = (b − a)2 . 12 Exponential: f (x) = λe−λx if x ≥ 0 0 otherwise. E(X) = 1/λ, Var(X) = 1/λ2 . Normal (Gaussian) with mean µ and variance σ 2 : Z x (x−µ)2 (t−µ)2 1 1 − 2 f (x) = √ e 2σ and F (x) = √ e− 2σ2 dt. σ 2π σ 2π −∞ Standard Normal (Gaussian) with µ = 0 and σ 2 = 1: Z z t2 1 − z2 1 f (z) = √ e 2 , and F (z) = Φ(z) = √ e− 2 dt. 2π 2π −∞ STAT/MTHE 353 – Final Exam, April 21, 2012 Page 12 of 12 The distribution function, Φ(z), of a standard normal random variable Note: Φ(−z) = 1 − Φ(z) for any z ∈ R.