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2. (12 marks] Let X1, X2, ..., X, be a random sample with the common density function f(x,0) = 0-4e-2/0 X>0 where e > 0 is an unknown parameter. = i a) [2 marks] Provide justification that the statistic X = n-1=1 Xi is a complete and sufficient statistic for . b) [1 mark] Determine the UMVUE of 0. c) (1 mark] Determine the MLE of 0. d) (1 mark] The density can also be written in the form EN N f(x,T) = te-Ta, > 0 = > where t = > 0 is an unknown parameter. Determine the MLE of T. e) [3 marks] Show that the UMVUE of 7 is given by T n - 1 Tumvue nX Hence, as usual the reciprocal of the MLE is the MLE of 1/0, but, in this situation, the reciprocal of the UMVUE is not the UMVUE of 1/0. Hint: consider E(X-1) and note that I(n) = (n − 1)T(n − 1) and n Τ = ΣΧ; Gamma(n,n). i=1 f) [2 marks] Determine the MLE of h(0) = P(X < 2). g) [2 marks] Determine the asymptotic distribution of h(0) = P(X < 2).

2. (12 marks] Let X1, X2, ..., X, be a random sample with the common density function f(x,0) = 0-4e-2/0 X>0 where e > 0 is an unknown parameter. = i a) [2 marks] Provide justification that the statistic X = n-1=1 Xi is a complete and sufficient statistic for . b) [1 mark] Determine the UMVUE of 0. c) (1 mark] Determine the MLE of 0. d) (1 mark] The density can also be written in the form EN N f(x,T) = te-Ta, > 0 = > where t = > 0 is an unknown parameter. Determine the MLE of T. e) [3 marks] Show that the UMVUE of 7 is given by T n - 1 Tumvue nX Hence, as usual the reciprocal of the MLE is the MLE of 1/0, but, in this situation, the reciprocal of the UMVUE is not the UMVUE of 1/0. Hint: consider E(X-1) and note that I(n) = (n − 1)T(n − 1) and n Τ = ΣΧ; Gamma(n,n). i=1 f) [2 marks] Determine the MLE of h(0) = P(X < 2). g) [2 marks] Determine the asymptotic distribution of h(0) = P(X < 2).