We wish to estimate:

var(X.bar) = mp(1 - p)/n.

Where S=X1+X2+...+Xn is a random sample from a binomial (m, p) population

1. Find an unbiased estimator of mp(1 - p)/n.

Note that since E[X1] = mp, X1/m estimates p well, so we may try X1 (m - X1)/m2 as an estimator of p(1 - p), and a constant multiple of X1(m - X1) as an estimator of mp(1 - p)/n.

2. Find the optimal estimator using the Rao-Blackwell Theorem.