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  "body": "\n## Definition\n\nLet $F \\colon \\mathbb{R} \\to [0, 1]$ be the [[cumulative distribution function|0MXMS3AE13]] of a real random variable $X$.\nThe quantile function of $X$ is the function $Q \\colon (0, 1) \\to \\mathbb{R}$ with\n$$Q(p) = \\inf \\{x \\in \\mathbb{R} : F(x) \\ge p\\}.$$\nHere $\\inf$ is the [[infimum|04KFTA34C5]]. The number $Q(p)$ is the $p$-quantile of $X$.\n\n## Remarks\n\nThe set in the definition is nonempty and bounded below for $0 \u003c p \u003c 1$, and the infimum is a minimum because $F$ is right-continuous. So $F(Q(p)) \\ge p$, and $Q(p) \\le x$ if and only if $p \\le F(x)$.\nIf $F$ is continuous and strictly increasing, then $Q = F^{-1}$.\nIf $U$ is uniform on $(0, 1)$, then $Q(U)$ has the cumulative distribution function $F$. This is inverse transform sampling.\n$Q(1/2)$ is a [[median|08W2CW8PKW]] of $X$, and $Q(1/4)$ and $Q(3/4)$ are the lower and upper quartiles.\n\n## Explanations\n\n### L1 human\n\nThe quantile function answers the question \"below which value does a fraction $p$ of the probability lie?\".\nFor the exponential distribution with rate $1$, $F(x) = 1 - e^{-x}$, so $Q(p) = -\\ln(1 - p)$.\n",
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