Now let's use Bayes' theorem and the binomial distribution toaddress a Bayesian inference question. You toss a bent coin Ntimes, obtaining a sequence of heads and tails. The coin has anunknown bias f of coming up heads. (a) If NH heads have occurred inN tosses, what is the probability distribution of f? Assume auniform prior P(f) = 1 and make use of the following result:integral 0 to 1 f^a (1 - f)^b df = a!b! / (a + b + 1)! (b) Sketch(or plot) the shape of the probability distribution of f for N = 5and NH = 2. (c) Now derive a formula for the most probable value off (the most probable value of f, denoted f ', is the value of fthat maximizes the probability distribution in (a)). What is f 'for N = 5 and NH = 2. Hint: maximize log P(f | NH, N) rather thanP(f | NH, N).
