2. A weighted coin has probability p for Heads and 1-p for Tails when it is flipped. We saw in class that if the prior d

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2 A Weighted Coin Has Probability P For Heads And 1 P For Tails When It Is Flipped We Saw In Class That If The Prior D 1 (120.32 KiB) Viewed 43 times

2. A weighted coin has probability p for Heads and 1-p for Tails when it is flipped. We saw in class that if the prior distribution for p is Beta(a,b), and we flip the coin n times with a result of k Heads and n-k Tails, then the posterior distribution for p is Beta(a + k, B+n-k). Recall that the Beta(a,b) pdf is T(a + B) f(x) = 22-1(1 - 0)8-1 r(a)T(B) (0 < x < 1) and the mean of the Beta(a, b) distribution is a/(a+B). (a) Suppose that the prior distribution for p is Beta(1,1), which is the same as Uniform(0,1). The coin is flipped 100 times, with a result of 60 Heads and 40 Tails. Write down the mean of the posterior distribution. If a is the .025 quantile of the posterior distribution, write down an integral equation for a (in which a appears as one of the limits of integration). Similarly, write down an integral equation for b, the .975 quantile of the posterior distribution. The value of a can be computed using the R command qbeta(.025, 61,41) or the Wolfram Alpha command InverseBetaRegularized (.025,61,41). Write 1 down the values of a and b to three decimal places. What is the posterior probability P(a <p <b)? (b) With a uniform prior for p as in part (a), suppose the coin is flipped 1000 times with 600 Heads and 400 Tails. Write down the mean of the posterior distribution and the interval estimate [a, b] for p, where a, b are computed as above. Now suppose instead that the coin is flipped 10000 times with 6000 Heads and 4000 Tails. Again, write down the mean of the posterior distribution and the interval estimate [a, b] for p.