Empirical distribution function

Remarks

For each $\omega$, $\hat{F}_n$ is the distribution function of the empirical measure $\frac{1}{n}\sum_{i} \delta_{X_i(\omega)}$, so it is non-decreasing, right-continuous, and has limits $0$ and $1$ at $-\infty$ and $+\infty$. Example: for the data $1, 2, 2, 5$, one has $\hat{F}_4(2) = 3/4$, because three of the four values are at most $2$. Non-example: the function $t \mapsto \frac{1}{n}\#\set{i : X_i < t}$ uses strict inequality; it is left-continuous and is not a distribution function in the usual convention. For fixed $t$ and an i.i.d. sample, $n \hat{F}_n(t)$ has the binomial distribution with parameters $n$ and $F(t)$. So $\hat{F}_n(t)$ is an unbiased estimator of $F(t)$ with variance $F(t)(1 - F(t))/n$.