Let X, X2,... be iid Bernoulli(p) random variables, that is, P(X = 1) = p and...

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Let X₁, X2,... be iid Bernoulli(p) random variables, that is, P(X₁ = 1) = p and P(X₁ = 0) = 1 - p. (a) Show that the variance of X, attains the Cramer-Rao Lower Bound, and hence X, is the best unbiased estimator for p. (b) For n ≥ 4, show that the product X₁ X₂ X3X₁ is an unbiased estimator of p¹, and use this fact to find the best unbiased estimator of p¹. (c) Let D₁ = (x₁=1, X₁+1=1) i=1 where IA 1 if the condition A holds and 0 otherwise. - Find E(D₁) and Var(D₁). - Show that → p in probability as n →∞o. - It is known that √n (D → N(0, p²-p¹) in distribution as n→→ ∞0. Let X₁, X2,... be iid Bernoulli(p) random variables, that is, P(X₁ = 1) = p and P(X₁ = 0) = 1 - p. (a) Show that the variance of X, attains the Cramer-Rao Lower Bound, and hence X, is the best unbiased estimator for p. (b) For n ≥ 4, show that the product X₁ X₂ X3X₁ is an unbiased estimator of p¹, and use this fact to find the best unbiased estimator of p¹. (c) Let D₁ = (x₁=1, X₁+1=1) i=1 where IA 1 if the condition A holds and 0 otherwise. - Find E(D₁) and Var(D₁). - Show that → p in probability as n →∞o. - It is known that √n (D → N(0, p²-p¹) in distribution as n→→ ∞0.