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STAT_333_Test_2_Solutions

STAT_333_Test_2_Solutions - STAT 333 Winter 2010 Test 2...

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STAT 333 Winter 2010 Test 2 SOLUTIONS Thurs, March 18, 4:00 5:30 pm First (given) name:__________________ Last (family) name:____________________ Student ID #:__________________ UW userid:____________________ Instructions: 1. Please fill in the above information 2. This test has 7 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 9 3 8 4 20 5 11 Total 60
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1. Suppose a monkey randomly and independently hits the white notes on a piano repeatedly, one at a time. The notes are {A, B, C, D, E, F, G}. [3] a) Find the expected # trials until the monkey first plays the melody “C D C D E D C D C”. Delayed renewal event with maximum overlap “C D C” So T CDCDEDCDC = T CDC + T CDCDEDCDC-bar But “C D C” is delayed renewal too with maximum overlap “C” So T CDCDEDCDC = T C + T CDC-bar + T CDCDEDCDC-bar = 7 + 7 3 + 7 9 using the Renewal Theorem = 40,353,957 [4] b) Derive the pgf F CDC (s) of the waiting time until the monkey first plays “C D C”. Delayed renewal sequence: d 0 = 0, d 1 = d 2 = 0, d n = (1/7) 3 , for n ≥ 3. Associated renewal sequence: 0 r = 1, 1 r = 0, 2 r = (1/7) 2 , n r = (1/7) 3 for n ≥ 3. So D CDC (s) = [s 3 /343]/(1 s) R CDC-bar (s) = 1 + s 2 /49 + [s 3 /343]/(1 s) = [1 s + (1 s)s 2 /49 + s 3 /343]/(1 s) So F CDC (s) = [s 3 /343]/[1 s + (1 s)s 2 /49 + s 3 /343] [3] c) Find the probability that “C D C” occurs for the first time on trial 6. (Hint: you can do this without expanding F CDC (s)) We want f 6 = P(“C D C” occurs for the first time on trial 6) We need _ _ _ C D C but not C D C C D C since in that case it has occurred first on trial 3, and not _ C D C D C, since in that case it has occurred first on trial 4. f 6 = P(“CDC” occurs on trial 6)–P(“CDC” occurs on 6 and 3)–P(“CDC” occurs on 6 and 4) = (1/7) 3 (1/7) 6 (1/7) 5 = 0.002847 [2] d) What two things are required for a delayed renewal event to occur infinitely often? The event must occur at least once with probability 1, and The associated renewal event must be recurrent (occur infinitely often)
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2. A fair six- sided die is rolled repeatedly. Let λ be the event “the m inimum roll so far is 3 [3] a) Explain why λ is a renewal event by comparing T λ and T λ
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