The probability of heads of a given coin is known to be

Chapter , Problem 8

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The probability of heads of a given coin is known to be either qo (hypothesis Ho) or ql (hypothesis HI ). We toss the coin repeatedly and independently, and record the number of heads before a tail is observed for the first time. We assume that 0 < qo < ql < 1, and that we are given prior probabilities P(Ho) and P(HI). For parts (a) and (b) , we also assume that P(Ho) = P(HI ) = 1/2. (a) Calculate the probability that hypothesis HI is true, given that there were exactly k heads before the first tail. (b) Consider the decision rule that decides in favor of hypothesis HI if k k* , where k* is some nonnegative integer, and decides in favor of hypothesis Ho otherwise. Give a formula for the probability of error in terms of k*, qo, and ql . For what value of k* is the probability of error minimized? Is there another type of decision rule that would lead to an even lower probability of error? (c) Assume that qo = 0.3, ql = 0.7, and P(HI) > 0.7. How does the optimal choice of k* (the one that minimizes the probability of error) change as P(HI ) increases from 0.7 to 1.0?