Suppose you have n independent and identically distributed observations of the random variable X, where X follows a Bernoulli distribution. The observations are x1, x2, . . . , xn. Let E(X) = p.

(a) (1 point) What is the best linear unbiased estimator of E(X)?

(b) (5 points) Show that your estimator from part a is unbiased. To receive full credit, you must provide a justification for each step.

(c) (5 points) Show that your estimator has the minimum variance out of all linear unbiased estimators of E(X) for the case where n = 2. To receive full credit, you must provide a justification for each step.

(d) (1 point) What is the variance of the sample mean when you have n independent observations?