Question: 2. Suppose that two independent binomial random variables X, and X, are observed where X] has a Binomial(n; p) distribution and X2 has a Binomial(2n;

2. Suppose that two independent binomial random variables X, and X, are observed where X] has a Binomial(n; p) distribution and X2 has a Binomial(2n; p) distribution. You may assume that n is known, whereas p is an unknown parameter. Define two possible estimators of p P1 = 3n (Xi + X2) and P2 = 2n (X1 + 0.5X2). (a) Show that both of the estimators Pi and P are unbiased estimators of p. (b) Find Var( Pi) and Var( P2). (c) Show that both estimators are consistent estimators of p. (d) Show that Pi is the most efficient estimator among all unbiased estimators. (e) Derive the efficiency of the estimator P relative to Pi

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