B1. (a) State (but do not prove) the Neyman-Pearson lemma. [3] The random variable X has the following probability density function e f(z;0) = o where z > 0, 6 is a unknown parameter (with 6 > 0) and T'(d) = [;y*'edy. (b) Under the assumption that one observation of X is to be made, derive the most powerful test of size o (where 0 17 Justify your answer. (3] Now suppose that a Bayesian hypothesis test of Hp: 6 =1 against H;: 6 =2 is planned. The decisions dy and d; correspond to choosing Hp and H;, respectively. The losses incurred from taking each of these decisions, where the value of the parameter of interest is 6, are L(0,do) =12(0 1); L(0,d,) = 18(2 ). (e) If the prior probabilities of the events {'Hg is true'} and {*H, is true'} are 3 and 2, respectively, and one observation of X is made, say x, determine the range of values of x for which H; would be chosen over Hy. (5]