5.pdf - STATISTICS 244 Due Thursday Problem Set 5...

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STATISTICS 244 Problem Set 5 9:0 0-10:20 TTh Due Thursday February 16, 2017 1. Many non-normal distributions are approximately normal, N( μ, s 2 ). One example is the Beta( a , b ) distribution, as long as a and b are moderately large and the Beta expectation and variance are used for μ and s 2 . Suppose X has a Beta density. (i) Use the normal approximation to find all three of P(|X–.6| <.001), P(|X–.6| <.01), and P(|X–.6| <.1), when X is (a) Beta(3,2), (b) Beta(30,20), (c) Beta(300, 200). (Present the results in a 3x3 table.) (ii) For case (a) only, find the same three probabilities exactly directly from the beta density by integration. 2. Election ties. What is the chance that there will be an exact tie in an election? Suppose there are n voters and that n is even (otherwise a tie is impossible). Suppose the voters vote independently of one another, each with probability p of voting for candidate A and probability 1 – p of voting for candidate B. Let X = total vote for A; then there will be a tie only if X =n/2. (a) What is the probability of a tie if p = .5 and n = 20? (b) What is the probability of a tie if p = .6 and n = 20? (c) Suppose p is unknown, and our uncertainty about it is described adequately by a Beta (4,4). What is the probability of a tie (i.e. the marginal probability P(X = n/2)), if n = 20?
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