1. (Risk minimization and a simplified binary classifier) In lecture, we learnt about the Neyman-Pearson framework: we seek to find a "best" critical region that maximizes the power of the test while main- taining a given significance level. Another major paradigm in the literature to define a "best" critical region is known as risk minimization that we now introduce. Let X ~N(д, 1). We are interested in testing Họ:g=0, Hi:p=1. As we only have a single sample X, we intend to use the test statistic T and critical region C to be respectively T=X, C={X> c}, where c> 0 is a constant. Let be the standard normal cdf. (a) Prove that the probability of type I error is a(c)-1-(c). The bracket of c on the left hand side is to indicate the dependence on c of a, that is, a is a function of c. (b) Prove that the probability of type II error is B(c) (c-1). The bracket of c on the left hand side is to indicate the dependence on c of ẞ, that is, ẞ is a function of c. (c) Define the risk R to be the sum of the probability of type I and type II error, that is, R(c) = a(c)+3(c) = 1 − (c) + (c− 1). In risk minimization, we seek to find an optimal critical region by minimizing R(c) with respect to c. Let c* be the resulting minimizer, that is, Prove that c* = arg min R(c). and hence = R(c) 24(-1/2).