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5. (a) Let / and V be independent standard normal random variables. The density func- tion of U is fu(u) = 2x and similarly for V. Define X = / and Y = p/ + v1 - p-V, where -1 0 otherwise The distribution of X is Gamma(a,#), which has density function fx(x) = nojolene for 1 > 0 otherwise where a > 1 and # > 0. i. Use the law of iterated expectations to find E(Y). [5 marks] ii. Work out the density of Y. [5 marks] 6. Let X1,.... X, be a random sample from the Uniform distribution on (0, #). Based on this sample, we want to perform a hypothesis test of Ho : 0 = 0, versus H1 : 0 > do. Our decision rule is to reject Ho in favour of Hi if X(m) > k, where Xin) is the sample maximum, and k is a specified positive constant. (a) Show that the probability density function of X(n) is (X(m(x) = na" -/0" for05150 otherwise. [4 marks] (b) If # = 1/2 and the probability of a Type I error is a, find an expression for k as a function of n and a. [7 marks] (c) If the true value of # is 3/4, find the smallest value of n required so that the power of this test is at least 0.98 when a = 0.05 and 6% = 1/2. [9 marks]