Empirical cumulative distribution function

Given a random sample ${𝑋 𝑗} 𝑛 𝑗 = 1$ of independent and identically distributed real random variables with CDF $𝐹$ , let $𝑅 𝑛 (𝑥)$ count how many of ${𝑋 𝑗} 𝑛 𝑗 = 1$ are less than or equal to $𝑥$ ; i.e.

𝑅 𝑛 (𝑥) = 𝑛 \sum 𝑗 = 1 1 {𝑋 𝑗 \leq 𝑥}

implying $𝑅 𝑛 (𝑥) \sim B i n (𝑛, 𝐹 (𝑥))$ . The empirical cumulative distribution function of ${𝑋 𝑗} 𝑛 𝑗 = 1$ is prob

𝐹 𝑛 (𝑥) = 1 𝑛 𝑅 𝑛 (𝑥) = 𝑛 \sum 𝑗 = 1 1 {𝑋 𝑗 \leq 𝑥}

and $𝐹 𝑛 (𝑥)$ converges almost surely to $𝐹 (𝑥)$ as $𝑛 \to \infty$ , hence it is an estimator of the true CDF.

Proof

proof By Kolmogorov’s law.