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  "body": "\n## Definition\n\nLet $X$ be a real [[random variable|0DKRR22AEQ]] on a probability space $(\\Omega, \\mathcal{F}, P)$.\n$X$ has a density, or is absolutely continuous, if there is a Borel measurable function $f \\colon \\mathbb{R} \\to [0, \\infty)$ with the following property. For each Borel set $B \\subseteq \\mathbb{R}$,\n$$P(X \\in B) = \\int_B f(x) \\, dx,$$\nwhere the integral is the Lebesgue integral.\n\n## Remarks\n\nSuch a function $f$ is a [[probability density function|04YG690N5V]] of $X$.\nIf $X$ has a density, then $P(X = x) = 0$ for each $x$, and the cumulative distribution function $F_X(x) = \\int_{-\\infty}^x f(t) \\, dt$ is continuous.\nA continuous cumulative distribution function does not imply a density. The Cantor distribution has a continuous cumulative distribution function and no density.\nThe same definition with $\\mathbb{R}^n$ and the Lebesgue measure on $\\mathbb{R}^n$ gives random vectors with a density.\n\n## Explanations\n\n### L1 human\n\nFor a random variable with a density, the probability of an interval is an area under a curve.\nSingle points have probability zero, just as a single point has no length.\n",
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