Suppose we choose a random sample of size n from a Bernoulli distribution with pdf f X (x)=p x (1?p) 1?x for x = 0, 1. Note that E( X )=p and Var ( X )=p(1?p) . Suppose we wish to find an estimator of the variance. Consider T=X ( X ?1) .

a) (3) Determine E(T ) . Hint: note that T can be written as X 2 - X and use what you know about the mean and variance of a sample mean.

b) (1) Since T is not unbiased for p(1?p) , how would you adjust T to construct an unbiased estimator?

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