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W 4207/5207 Name: Spring 2020 UNI: Final Exam 05/11/2020 Time Limit: 180 Minutes This exam contains 8 pages (including this cover page) and 7 questions. Total of points is 100 Instructions: Do any 4 problems. If you do more than 4, only 4 will be graded. If you do more than 4, clearly indicate the problems you want graded in the table at the end of this page. You may use your notes, textbook, and calculator. Work alone without assistance from anyone else either in person, by telephone or by email. Do not use a computer, except to receive and hand in the exam. Each problem has 5 parts, each worth 5 points, thus each problem is worth 25 points. Upload your submission to Courseworks (within the “Assignments” section) before 12:15. Please note that Courseworks will only allow submissions in PDF format. Do not email the solution to the Professor nor the TAs. Grade Table Question Points Score 25 25 25 25 Total 100 W 4207/5207 Final Exam - Page 2 of 8 05/11/2020 1. (25 points) A fair die is rolled until the first even outcome, (2, 4, 6), is obtained. Let N equal the number of rolls required. For example, if we observe (1, 3, 3, 1, 5, 1, 5, 3, 6, . . . ), then N = 9. Find (a) (5 points) E (N |SN = 9) (b) (5 points) E [ (0.8)N ] (c) (5 points) E (SN |N) (d) (5 points) E(SN) (e) (5 points) Var(SN) W 4207/5207 Final Exam - Page 3 of 8 05/11/2020 2. (25 points) Consider {Xi, i ≥ i} iid Exponential(2), {Yi, i ≥ i} iid Exponential(3), {N1(t), t ≥ 0} a Poisson process with intensity 1, and {N2(t), t ≥ 0} a Poisson pro- cess with intensity 2. Assume that {Xi}, {Yi}, {N1(t), t ≥ 0} and {N2(t), t ≥ 0} are independent. Define W1(t) = N1(t)∑ i=1 Xi, W2(t) = N2(t)∑ i=1 Yi, {N(t) = N1(t) +N2(t), t ≥ 0}, {W (t) = W1(t) +W2(t), t ≥ 0} (a) (5 points) Is {W (t), t ≥ 0} a compound Poisson Process? Why or why not? (b) (5 points) Find P (W1(3) > W2(3)|N1(3) = 2, N2(3) = 3) (c) (5 points) Define M = {min k : ∑k i=1Xi > 3}. Find E (∑M i=1Xi ) (d) (5 points) Define T3 to be the time of the 3rd event for the process {N(t), t ≥ 0} defined above. Find P (T3 > 1.5) (e) (5 points) Let N ∼ Bin(10, 0.3) with N independent of {Yi}. Find, E ( min 1≤i≤N+1 Yi ) W 4207/5207 Final Exam - Page 4 of 8 05/11/2020 3. (25 points) Consider a standard Brownian motion process {X(t), t ≥ 0}. Find (a) (5 points) P (X(1) < 1="" +x(3)).="" (b)="" (5="" points)="" var="" (="" sup="" 2≤t≤5="" x(t)="" )="" .="" (c)="" (5="" points)="" let="" y="" denote="" the="" first="" zero="" for="" the="" brownian="" motion="" process="" after="" time="" t="5." find="" p="" (y=""> 8). (d) (5 points) Define X̃(t) = 3X(t) + 2t. Let T denote the first time that X̃(t) = √ 2, thus T = {inf t : X̃(t) = √ 2}. Find the mean and variance of T . (e) (5 points) Find P ( X21 + (X2 −X1) 2 + (X3 −X2)2 + (X4 −X3)2 X21 + (X2 −X1) 2 > 2 ) W 4207/5207 Final Exam - Page 5 of 8 05/11/2020 4. (25 points) Consider a Markov chain with state space {0, 1, 2, 3} and transition matrix P = 0 1 2 3 0 0 0 0 1 1 0 0 1 0 2 1/4 1/4 1/4 1/4 3 0 1 0 0 (a) (5 points) Find lim k→∞ E [ (X(k))X(k) |X(0) = 2 ] (b) (5 points) Define A = {1, 3}. Find the expected number of steps to go from 2 to A for the first time, that is find, ET2,A = E [ min k Xk ∈ A|X(0) = 2 ] (c) (5 points) Find P (visit 0 before 3|X(0) = 2) (d) (5 points) Find P 4 (0, 0) = P (X(4) = 0|X(0) = 0) (e) (5 points) Find lim n→∞ P (Xn+1 = 1, X2n = 2|X(0) = 3) W 4207/5207 Final Exam - Page 6 of 8 05/11/2020 5. (25 points) (a) (5 points) Consider a continuous time Markov chain with state space {0, 1, 2} and infinitesimal matrix Q = 0 1 2 0 -4 2 2 1 3 -4 1 2 3 1 -4 Find an explicit expression for P (t) 01 = P (X(t) = 1|X(0) = 0) (b) (5 points) A factory has 2 machines and 2 repairmen. The operating time of a ma- chine is exponential with mean 1 hour. The repair time of a machine is exponential with mean 15 minutes. A busy period begins when 1 of 2 working machines fail and ends the next time instant that both are working. Find the expected length of a busy period. (c) (5 points) We have 6 balls distributed among 2 urns, A and B. At each stage one of the urns is randomly chosen. If the chosen urn is non-empty then one of its balls is moved to the other urn. Otherwise, we do nothing. Find the long term proportion of time that one of the urns is empty. (d) (5 points) Consider a continuous time MC with state space S = {1, 2, 3} and in- finitesimal matrix Q = −3 1 22 −4 2 1 4 −5 Find the stationary distribution. (e) (5 points) In (d), find P (reach 2 before 1|X0 = 3) W 4207/5207 Final Exam - Page 7 of 8 05/11/2020 6. (25 points) Consider a branching process with offspring distribution Z ∼ Bin(3, 0.7). Let Xn denote the population at time n, with X0 = 1. Find, (a) (5 points) P (X1 = 2|X2 = 5) (b) (5 points) P (X2 = 0|X3 = 0) (c) (5 points) Var (X4|X2 = 3) (d) (5 points) lim n→∞ P (Xn = 0) (e) (5 points) P (X4 ≥ 1|X3 ≥ 1) W 4207/5207 Final Exam - Page 8 of 8 05/11/2020 7. (25 points) (a) (5 points) Consider 3 independent Poisson process with respective intensities 1, 2, 3. Define W to be the first time that all 3 processes each have at least one event occurring. Thus W = min{t : Ni(t) ≥ 1, i = 1, 2, 3}. Find E(W ) (b) (5 points) Consider a Markov chain with P = 0 1 2 3 0 1 0 0 0 1 0.3 0.5 0.2 0 2 0 0.2 0.3 0.5 3 0 0 0 1 Find the probability that starting in state 1, that the chain eventually gets absorbed into state 3. (c) (5 points) Claims arrive to an insurance company according to a Poisson process with rate λ = 5 claims per day. The claim sizes are iid with pdf f(x) = (0.003)2xe−0.003x, x > 0 Find the probability that there will be at least 4 claims of size $1000 or greater over a 3 day period (d) (5 points) Suppose that X, Y, Z are independent exponential random variables with E(X) = 1, E(Y ) = 2 and E(Z) = 3 (Thus X ∼ Expo(1), Y ∼ Expo(1/2), Z ∼ Expo(1/3). Find E ( 1 6X + 3Y + 2Z ) (e) (5 points) Consider a M/G/∞ queue in which service times have pdf, f(x) = { 1 2 , 0 < x="">< 2="" 0,="" elsewhere="" the="" arrivals="" are="" from="" a="" poisson="" process="" with="" λ="5." find="" the="" distribution="" of="" the="" number="" of="" customers="" in="" service="" at="" time="" 3,="" who="" arrived="" to="" the="" system="" during="" 2.1="">< t="">< 2.7>