Following a recent Climate Action Plan (CAP) conference, a particular model was agreed for measuring global earth tremors. A series of n positive measurements are to be taken, which are assumed to be independent observations of a random variable that are Uniformly distributed on (0, v), where v> 0. A Climate Risk Actuary adopts the model agreed by the CAP conference. The Actuary knows only that the number, R, of the measurements that are less than 1 is r, with the remaining n-r being greater than 1. (i) (a) (b) (d) Show that the probability for a single measurement to be less than 1 is [2] Show that the maximum likelihood estimate of v is == Identify which one of the following expressions gives the Cramer-Rao lower bound for estimating v. A B D v(1-v) 71 2(1-1)2 n (x-1) 1(x-1) n [5] [3] Write down the asymptotic distribution of , using your answer from part (i)(c). [2] W lower bound for estimating v. v(1-v) (d) A B C D n (1-v) n1 11 v2(x-1) Write down the asymptotic distribution of , using your answer from part (i)(c). [2] In a random sample of 500 measurements, exactly 75 measurements are less than 1. (ii) (a) Calculate an estimate for the standard error of 9. [2] (b) Determine an approximate two-sided 99% confidence interval for v. [2] [3] Perform a test for the following hypotheses, using your asymptotic distribution of from part (i)(d): Ho: v= 10 vs Hv