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  3. please show your works thank you...

Question: please show your works thank you...

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(3) [30 pts] Consider the 2000 U.S. presidential election of Bush v. Gore, the outcome of which was decided by a tiny margin in the state of Florida. Let μ denote the true fraction of votes won by Bush in Florida. In any count i of the votes, there is some error (denoted by j), due to human mistakes or errors in the way the machine reads the ballots. The error component for a given count of the ballots i is denoted є, which is distributed according to Unif(-.05, +.05). Note that this implies Eei 0. The error term is i.i.d. across any count of the ballots. Write the outcome of any count: (For this question, assume Bush and Gore are the only two candidates, and that Bush is declared the victor if he gets at least .5 of the votes.) (a) Suppose Bush truly won 49% of the votes, that is, -,49. What is the probability that Gore is declared the winner of the election if the votes are counted once? What is the probability that Bush is declared the winner if the votes are counted once? Hint: the pdf of a uniformly distributed random variable X on the interval la, b) is f(x) = b-a for a 〈 x 〈 b and 0 otherwise. (b) Florida state law mandates a recount of the votes if the first count is really close. According to law, the votes will be recounted once and the outcome of that recount will determine the election result. A Fox News commentator argues that there is no point to recounting the votes because there is just as much error in the recount as there is in the initial count. (i) You propose a novel idea, which is to use the sample mean of the two counts as an estimator μ for μ using your data on the two counts: {Vi}2-1. Show that your estimator is unbiased: i.e. thatE (ii) Show that the variance of your estimator for μ is smaller than the variance of the one that follows Florida state law (i.e. Florida state law uses the recount as its estimator: μ U2). Hence you are looking to prove that var(ngta) var(y2) Note: it is not necessary to calculate the variance of a uniform random variable to solve this question. Simply designate var(ci-ơ2

PLEASE SHOW YOUR WORKS, THANK YOU.

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