Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables following the distribution \(F\). Prove that for a fixed value of \(t \in \mathbb{R}\), the empirical distribution function \(\hat{F}_{n}(t)\) is an unbiased estimator of \(F(t)\) with standard error \(n^{-1 / 2}\{F(t)[1-F(t)]\}^{1 / 2}\).