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STAT_333_Test_2_Solutions

STAT_333_Test_2_Solutions - STAT 333 Spring 2008 Term Test...

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STAT 333 Spring 2008 Term Test 2 SOLUTIONS Wed, July 9 4:30 – 6:00 pm First (given) name:__________________ Last (family) name:____________________ Student ID #:__________________ UW userid:____________________ Instructions: 1. Please fill in the above information 2. This test has 6 pages, including this cover page 3. Answer all questions in the space provided 4. You have 90 minutes for the test 5. Show all your work and justify your steps 6. Good luck! Question Marks available Marks obtained 1 12 2 8 3 8 4 12 5 10 6 (BONUS) 4 Total 50
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1. Suppose an infinite sequence of letters are selected randomly from the set {A, B, C, D, R} [3] a) Find the expected # of trials until the first occurrence of the word “ABRACADABRA” Delayed renewal event with maximum overlap “ABRA” So T ABRACADABRA = T ABRA + A ABRACADABR T But “ABRA” is delayed renewal too with maximum overlap “A” So T ABRACADABRA = T A + ABRA T + A ABRACADABR T = 5 + 5 4 + 5 11 using the Renewal Theorem = 48,828,755 [4] b) Derive the pgf F(s) of the waiting time until we first observe “ABRA” Delayed renewal sequence: d 0 = 0, d 1 = d 2 = d 3 = 0, d n = (1/5) 4 , for n 4. Associated renewal sequence: 0 r = 1, 1 r = 2 r = 0, 3 r = (1/5) 3 , n r = (1/5)4 for n 4. So D ABRA (s) = [s 4 /625]/(1 – s) ABRA R (s) = 1 + s 3 /125 + [s 4 /625]/(1 – s) = [1 – s + (1 – s)s 3 /125 + s 4 /625]/(1 – s) So F ABRA (s) = [s 4 /625]/[1 – s + (1 – s)s 3 /125 + s 4 /625] [3] c) Find the probability that “ABRA” occurs for the first time on trial 7. (Hint: you can do this without expanding F(s)) We want f 7 = P(“ABRA” occurs for the first time on trial 7) We need _ _ _ A B R A but not A B R A B R A since in that case it has occurred first on 4. So f 7 = P(“ABRA” occurs on trial 7) – P(“ABRA” occurs on 7 and 4) = (1/5) 4 – (1/5) 7 = 0.0015872 [2] d) What two things are required for a delayed renewal event to occur infinitely often?
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