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2. [16 marks] Let X1. .... X,, and Y1, ..., Y,, be two independent random samples and suppose that X, # N(0, o') and Y, # N(0, 20?), where the N(u, ") probability density function is f ( 2;14, 02 ) = -exp _ (2-14)2 ] 202 Define the following quantities: and Ty - ELY? n Also define two possible estimators of o' S = Txt Ty and $; - - (7x + TY). You may use the fact that if Z has a standard normal distribution then Var(Z?) = 2. (a) Give the distribution of . (b) Show that the following are all true: Var ( x ? ) = E ( x ? ) + CEC RS ) ? E(X? ) = 2 E( Y, ? ) = 202 Var(X?) = 20* Var(Y?) = 80*. (c) Show that $7 and S, are both unbiased estimators of a?. (d) Find Var(S?) and Var($4). (e) Derive the efficiency of the estimator S, relative to $2. (f) Show that both estimators are consistent estimators of of. (g) Which estimator would you prefer? Provide a reason