Question: B1 For n e N, let X1, X2, ..., X, be i.i.d. random variables each with pdf 72 f(x) = 2.1 e 0 for x

B1 For n e N, let X1, X2, ..., X, be i.i.d. random variables each with pdf 72 f(x) = 2.1 e 0 for x (0,00), where 0 (0,0) is an unknown parameter. (a) Show that 2+, X} is the MLE of 0. 7 [7] (b) Show that the sampling distribution of is Gam(n,n0-1). Hint: Compute the distribution of X{ first. You may also use the ad- ditive property of Gamma-distributed random variables given in the preamble. 6] (c) Hence or otherwise compute the bias and variance of . [2] (d) Compute the CRLB for 0. Does @ achieve the same mse as an unbiased estimator attaining the CRLB? Justify. [5] (e) Obtain an approximate 95% confidence interval for based on asymptotic normality of . [3] (f) Remembering that an interval (al, n), bl, n)] is a 95% confidence interval if and only if P([al,n), 6(@,n)] = 0) > 0.95, decide whether the interval computed in (e) actually is a 95% confidence interval in the case n = 2. Discuss your findings commenting on the statistical implications of using the approximate confidence in- terval in (e) assessing their severity. Hint: Using the sampling distribution of established in (b), you will need to calculate the (1+1.96) va S). You may require integration by parts and a pocket calculator. [8] probability P (1+1,96/? a

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