Given a data set {xi: 0 i < N} of N values, where each value xi is independent and drawn from a normal (Gaussian) distribution. The values have different (known) standard deviations, given by oi. Use Bayes' theorem to prove that the most probable value 0 for the mean of the set is the weighted average of the data. Assume a flat prior for the mean . b) Assume a coin with unknown bias H = [0... 1] is thrown N times (H = 0.5 for a fair coin, H = 1 for a coin that always throws heads). The number of times heads comes up is R. Start from Bayes' theory and derive the best estimate of the bias, HO, and its variance o 2 H, given R and N.