Question

MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e) 15] [10] distribution where θ > 0 is unknown. We reject the simple null hypothesis Ho : θ-2 in favour of the simple alternative Hi : θ 4 whenever the sample mean of the observations is greater than or equal to 3. Show that this is a most powerful test of Ho against Hi- [10] 9. (a) Let X and Y, be simple random variables (i) Show that E(XbY) aE(X) +bE(Y (ii) Show that if X-Y, then E(X) BY). for all a, bE R. [10] (b) Let X0: X1: X2, . . . be a sequence of independent and identically distributed random variables, and let N > 0 be the index of the first number in the sequence that exceeds the initial value Xo. Show that E(N) is infinite 15] (c) Let X be a discrete random variable with the following PMF. otherwise Show that E(X) is undefined 5/6

Answer 1

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