3. MLE and MAP Estimates The coin is tossed 10 times and the resulting sequence is HTTHHHTHTH. In the parts below, we refer to this information as "the data". a) Under the assumption that the coin lands heads with a fixed unknown probability p, find the MLE of p based on the data. b) Suppose now that the coin lands heads with a random probability X. Let the prior density of be uniform on the unit interval. Find the MAP estimate of the probability of heads, given the data. c) Show that if r > 1 and s > 1 then the mode of the beta (r, s) distribution is (r - 1)/(r+s - 2). Remember to ignore multiplicative constants and take the log before maximizing. d) Suppose instead that the prior density of X is f(x) = 4x if 0 < x < 1 and 0 otherwise. Find the MAP estimate of the probability of heads, given the data.