STA447H1_08SPRING1_853 - 8 51 3 STA447 STA2006 ID : CIRCLE...

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Unformatted text preview: 8 51 3 STA447 STA2006 ID : CIRCLE ONE: Given name(s): ad e r Surname: Student #: (February 28, 2008, 6:10 p.m. Pages: 6; Time: 130 minutes.) Total points: 65.) // Uo Up lo ut fT ad or S er on tu ID : to de 51 .s nt 38 Up tu B lo de ud ad er nt dy ID bu : dd 51 3 y (Questions: 6; .c om Up lo STA 447/2006S, Winter 2008: In-Class Test NO AIDS ALLOWED – NOT EVEN CALCULATORS. Up l oa d er ID : 51 38 ht tp : 8 1. [8 points] Let (pij ) be the transition probabilities for random walk on the graph whose vertices are V = {1, 2, 3, 4}, with a single edge between each of the four pairs (1,2), (2,3), ( n) (3,1), and (3,4), and no other edges. Compute (with full explanation) limn→∞ p13 . 1 This test is copyrighted by the uploader/owner with above Student Buddy ID. Unauthorized reproduction/distribution is strictly prohibited. Solution (if any) is NOT audited, so use at your discretion. 8 51 3 ID : 2. Consider the Markov chain with state space S = {1, 2, 3}, and transition probabilities given by p11 = 1/6, p12 = 1/3, p13 = 1/2, p22 = p33 = 1, and pij = 0 otherwise. .c om // Uo Up lo ut fT ad or S er on tu ID : to de 51 .s nt 38 Up tu B lo de ud ad er nt dy ID bu : dd 51 3 y Up lo ad e r (a) [4 points] Compute (with explanation) f12 (i.e., the probability, starting from 1, that the chain will eventually visit 2). ( n) [3 points] Prove that p12 ≥ 1/3, for any positive integer n. (c) [2 points] Compute 8 (b) ( n) p12 . ht tp : ∞ n=1 Up l oa d er ID : 51 38 (d) [3 points] Relate the answers in parts (a) and (c) to theorems from class about when ( n) ∞ fij = 1 and when n=1 pij = ∞. 2 This test is copyrighted by the uploader/owner with above Student Buddy ID. Unauthorized reproduction/distribution is strictly prohibited. Solution (if any) is NOT audited, so use at your discretion. 8 51 3 ID : 3. Let S = Z (the set of all integers), and let h : S → [0, 1] with i∈S h(i) = 1. Consider the transition probabilities on S given by pij = (1/4) min(1, h(j )/h(i)) if j = i−2, i−1, i+1, or i + 2, and pii = 1 − pi,i−2 − pi,i−1 − pi,i+1 − pi,i+2 , and pij = 0 whenever |j − i| ≥ 3. .c om ht tp : 8 // Uo Up lo ut fT ad or S er on tu ID : to de 51 .s nt 38 Up tu B lo de ud ad er nt dy ID bu : dd 51 3 y Up lo ad e r ( n) (a) [10 points] Assuming that h(i) > 0 for all i, prove that limn→∞ pij = h(j ) for all i, j ∈ S . (Carefully justify each step.) Up l oa d er ID : 51 38 (b) [5 points] Show by example that part (a) might be false if we do not assume that h(i) > 0 for all i. [For definiteness, we take min(1, h(j )/h(i)) ≡ 1 whenever h(i) = 0.] 3 This test is copyrighted by the uploader/owner with above Student Buddy ID. Unauthorized reproduction/distribution is strictly prohibited. Solution (if any) is NOT audited, so use at your discretion. 8 51 3 ID : 4. Consider a Markov chain {Xn } with state space S = {1, 2, 3, 4, 5}, X0 = 4, and transition probabilities specified by p11 = p55 = 1, p21 = 5/7, p24 = p25 = 1/7, p31 = p32 = p33 = p34 = p35 = 1/5, and p43 = p45 = 1/2. Let T = min{n ≥ 1 : Xn = 1 or 5}. [8 points] Determine (with full explanation) whether or not {Xn } is a martingale. (b) [4 points] Compute P(XT = 5). [Hint: part (a) might help.] .c om Up l oa d er ID : 51 38 ht tp : 8 // Uo Up lo ut fT ad or S er on tu ID : to de 51 .s nt 38 Up tu B lo de ud ad er nt dy ID bu : dd 51 3 y Up lo ad e r (a) 4 This test is copyrighted by the uploader/owner with above Student Buddy ID. Unauthorized reproduction/distribution is strictly prohibited. Solution (if any) is NOT audited, so use at your discretion. 8 51 3 ID : 5. Consider a Markov chain {Xn } on the state space S = {0, 1, 2, 3, . . .}, with X0 = 100, and pij = 1/(2i + 1) if 0 ≤ j ≤ 2i, otherwise pij = 0. .c om 8 // Uo Up lo ut fT ad or S er on tu ID : to de 51 .s nt 38 Up tu B lo de ud ad er nt dy ID bu : dd 51 3 y Up lo ad e r (a) [5 points] Prove that {Xn } is a martingale. (You may assume without proof that E|Xn | < ∞ for all n.) Up l oa d er ID : 51 38 ht tp : (b) [5 points] Prove that P(∃n ≥ 1 : Xn = 1000) < 1/6. [Hint: the martingale maximal inequality might help.] 5 This test is copyrighted by the uploader/owner with above Student Buddy ID. Unauthorized reproduction/distribution is strictly prohibited. Solution (if any) is NOT audited, so use at your discretion. 8 51 3 Let {N (t)}t≥0 be a Poisson process with rate λ > 0. ID : 6. [6 points] Compute the conditional probability qλ ≡ P(N (4) = 1 | N (5) = 3). // Uo Up lo ut fT ad or S er on tu ID : to de 51 .s nt 38 Up tu B lo de ud ad er nt dy ID bu : dd 51 3 y Up lo .c om ad e r (a) Up l oa d er ID : 51 38 ht tp : 8 (b) [2 points] Compute q2λ / qλ . (That is, determine the fraction by which the probability in part (a) changes if we replace λ by 2λ.) [END] 6 This test is copyrighted by the uploader/owner with above Student Buddy ID. Unauthorized reproduction/distribution is strictly prohibited. Solution (if any) is NOT audited, so use at your discretion. ...
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This document was uploaded on 03/27/2010.

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