Question: 1. (25 marks) Let X1, X2, ..., X, be a random sample from the Bernoulli distribution, say, P(X = 1) = 0, P(X = 0)

1. (25 marks) Let X1, X2, ..., X, be a random sample from the Bernoulli distribution, say, P(X = 1) = 0, P(X = 0) = 1 - 0. (a) (6 marks) Let S,, = >#-, Xi, find the distribution of S ,. (b) (6 marks) Show X(1 - X) is a biased estimator for 0(1 - 0), where X = ET-, X;. (c) (8 marks) Find the Cramer Rao Lower Bound for the variance of an unbiased estimator for 0(1 - 0). (d) (5 marks) Does the variance of any unbiased estimator for 0(1 -0) achieve this bound? Why? Explain in details

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