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  "body": "\n## Statement\n\nLet $F \\colon \\R \\to [0, 1]$ be a [[cumulative distribution function|0H8H7ZTHW2]], that is, non-decreasing, right-continuous, with limits $0$ at $-\\infty$ and $1$ at $+\\infty$.\nLet $F^{-1} \\colon (0, 1) \\to \\R$ be its generalized inverse, the [[quantile function|0X60VE98HN]]\n$$F^{-1}(u) = \\inf \\set{x \\in \\R : F(x) \\ge u}.$$\nLet $U$ be a [[random variable|0PNRKMRC55]] with the [[uniform distribution|0AREFQ9Z65]] on $(0, 1)$.\nThen the random variable $X = F^{-1}(U)$ has cumulative distribution function $F$.\n\n## Remarks\n\nThe key fact is that for $u \\in (0, 1)$ and $x \\in \\R$, $F^{-1}(u) \\le x$ if and only if $u \\le F(x)$; this uses right-continuity of $F$, which makes the infimum attained.\nHence $P(X \\le x) = P(U \\le F(x)) = F(x)$.\nExample: the [[exponential distribution|0E99F51GVB]] with rate $\\lambda \u003e 0$ has $F(x) = 1 - e^{-\\lambda x}$ for $x \\ge 0$, so $F^{-1}(u) = -\\ln(1 - u)/\\lambda$; since $1 - U$ is also uniform, $-\\ln(U)/\\lambda$ is exponential too.\nExample: for a discrete distribution with values $x_1 \u003c x_2 \u003c \\cdots$ and probabilities $p_1, p_2, \\ldots$, the method returns $x_k$ when $p_1 + \\cdots + p_{k-1} \u003c U \\le p_1 + \\cdots + p_k$.\nThe generalized inverse is needed: when $F$ has jumps or flat parts, the ordinary inverse does not exist.\nA converse holds when $F$ is continuous: then $F(X)$ is uniform on $(0, 1)$, the probability integral transform.\n\n## Explanations\n\n### L1 human\n\nA computer can produce uniform random numbers between $0$ and $1$.\nTo get random numbers with any other distribution, feed each uniform number into the inverse of the distribution function.\nValues of $u$ near $0$ give values from the lower tail, values near $1$ give the upper tail, and the shape of the distribution comes out exactly right.\nFor the exponential distribution this means $-\\ln(u)/\\lambda$.\n\n### L2 human\n\nThe method is exact and uses one uniform number per sample, so it is the first choice whenever $F^{-1}$ has a closed form or a fast accurate approximation.\nFor the normal distribution there is no closed form, and one uses numerical approximations of $F^{-1}$ or other methods such as the Box–Muller transform or the ziggurat algorithm.\nBeing monotone in $U$, it preserves the order of the uniform inputs, which is useful for antithetic variates, for quasi-Monte Carlo and for coupling arguments.\nIt also explains copulas: any multivariate distribution is obtained from its copula by applying the marginal quantile functions coordinatewise.\n",
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