Question: 1. (a) Let {X1, X2, ...; Xn} be a random sample of size n from the following probability density function: 1 fx (x; 0, 0)

1. (a) Let {X1, X2, ...; Xn} be a random sample of size n from the following probability density function: 1 fx (x; 0, 0) = "-le-x/0 (a - 1)!90 for > > 0, and 0 otherwise, where o > 0 is known, and 0 > 0. i. Derive the maximum likelihood estimator of 0. (You do not need to verify the solution is a maximum.) (7 marks) ii. Show that the estimator derived in part i. is mean square consistent for 0. Hint: You may use the fact that E(X ) = Q0 and Var(X ) = Q02. (6 marks)

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