Let F hat denote the empirical distribution function based on a random sample X1,....Xn with common cdf F.
Show that E[ F hat (x)] = F(x) and Var[ F hat (x)] = F(x)(1 - F(x))=n.
Hence show that F hat (x) is a consistent estimator of F(x) provided 0 < F(x) < 1.