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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be real random variables, typically [[i.i.d.|0AJN7WHBJZ]] with [[cumulative distribution function|0MXMS3AE13]] $F$.\nThe *empirical distribution function* of the sample is the random function $\\hat{F}_n \\colon \\R \\to [0, 1]$ given by\n$$\\hat{F}_n(t) = \\frac{1}{n} \\sum_{i=1}^{n} \\mathbf{1}\\set{X_i \\le t},$$\nwhere $\\mathbf{1}\\set{X_i \\le t}$ is $1$ if $X_i \\le t$ and $0$ otherwise.\n\n## Remarks\n\nFor 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 $+\\infty$.\nExample: for the data $1, 2, 2, 5$, one has $\\hat{F}_4(2) = 3/4$, because three of the four values are at most $2$.\nNon-example: the function $t \\mapsto \\frac{1}{n}\\#\\set{i : X_i \u003c t}$ uses strict inequality; it is left-continuous and is not a distribution function in the usual convention.\nFor fixed $t$ and an i.i.d. sample, $n \\hat{F}_n(t)$ has the binomial distribution with parameters $n$ and $F(t)$.\nSo $\\hat{F}_n(t)$ is an unbiased estimator of $F(t)$ with variance $F(t)(1 - F(t))/n$.\n",
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