7. Two hypotheses. The probability of heads of a given coin is known to be either qo (hypothesis Ho) or q (hypothesis H). 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 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 91. 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 = 0.3,91 0.7, and P(H1) > 0.7. How does the optimal choice of k* (the one that minimizes the probability of error) change as P(H) increases from 0.7 to 1.0? = 7. Two hypotheses. The probability of heads of a given coin is known to be either qo (hypothesis Ho) or q (hypothesis H). 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 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 91. 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 = 0.3,91 0.7, and P(H1) > 0.7. How does the optimal choice of k* (the one that minimizes the probability of error) change as P(H) increases from 0.7 to 1.0? =