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  "body": "\n## Definition\n\nLet $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\\Omega, \\mathcal{F}, P)$.\nThe cumulative distribution function of $X$ is the function $F_X \\colon \\mathbb{R} \\to [0, 1]$ with\n$$F_X(x) = P(X \\le x) \\quad \\text{for } x \\in \\mathbb{R}.$$\nMore generally, the cumulative distribution function of a [[probability distribution|0QB913WM5E]] $\\mu$ on $\\mathbb{R}$ is $F(x) = \\mu((-\\infty, x])$.\n\n## Remarks\n\nFor $a \u003c b$, $P(a \u003c X \\le b) = F_X(b) - F_X(a)$.\n$F_X$ is nondecreasing and right-continuous, with limit $0$ at $-\\infty$ and limit $1$ at $+\\infty$. Conversely, each function with these properties is the cumulative distribution function of some distribution.\nThe jump of $F_X$ at $x$ is $P(X = x) = F_X(x) - \\lim_{t \\to x^-} F_X(t)$.\nFor a fair die, $F_X(x) = \\lfloor x \\rfloor / 6$ for $0 \\le x \u003c 6$, $F_X(x) = 0$ for $x \u003c 1$, and $F_X(x) = 1$ for $x \\ge 6$.\n\n## Explanations\n\n### L1 human\n\nThe cumulative distribution function adds up the probability from the left up to the point $x$.\nIts graph starts at height $0$ on the far left, never goes down, and reaches height $1$ on the far right.\n",
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