1. We wish to estimate the probability of heads, denoted by 6, of a biased coin. We model 0 as the value of a random variable 9 that is uniformly distributed at [0, 1]. We consider n independent tosses and let X be the number of heads observed. (a) Find the PMF of X given 8, px'9(k|9). (b) Find the posterior PDF of 8 given X, petx(9|k). (Hint: You don't need to calculate px(k). Instead, use px(k) = l/c for some c independent of 6.) (c) Find the MAP estimator of 8, @MAP. k!(nk)! . . (mm In your calculatlon. (d) Find the LMS estimator of 9, (9mg. You may use the result of I; ma 6)\""'d6 =