324.final.2005.solution - Stat324 Final Exam Solutions Moo...

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Stat324: Final Exam SolutionsMoo K. Chung[email protected]May 11, 2005You may use the followingRoutput for the answering exam problems.> qnorm(c(0.005,0.01,0.025,0.05,0.1))[1] -2.575829 -2.326348 -1.959964 -1.644854 -1.281552> qt(0.025,5:11)[1] -2.57 -2.44 -2.36 -2.30 -2.26 -2.22 -2.20> qt(0.05,5:11)[1] -2.01 -1.94 -1.89 -1.85 -1.83 -1.81 -1.79> qt(0.01,5:11)[1] -3.36 -3.14 -2.99 -2.89 -2.82 -2.76 -2.71> qf(0.025,1,5:11)[1] 0.001084778 0.001067109 0.001054611 0.001045308 0.0010381150.001032389 [7] 0.001027723> qf(0.05,1,5:11)[1] 0.004344768 0.004273760 0.004223537 0.004186155 0.0041572560.004134252 [7] 0.004115506> qf(0.01,1,5:11)[1] 0.0001735012 0.0001706780 0.0001686808 0.0001671942 0.0001660447[6] 0.0001651297 0.00016438401. Two playersAandBflip a biased coin alternately and the first player to obtain a head wins.The probability ofobtaining a head isp >0at each toss. SupposeAflips the coin first.(a) Find the probability thatAwins the game. Derive your result [10pts].(b) Find the expected number of tossesAmust make tillAwins. Derive your result [10pts].Solution.
2.2. (a) Suppose thatXandYare Bernoulli random variables with parameter0< p <1such that cov(X, Y) = 0. ProvethatXandYare independent [10pts].(b) LetX1,· · ·, Xnbe a random sample from the Bernoulli distribution with parameterp. Find an unbiased estimatorfor(1-p)2[10pts].

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