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  "body": "\n## Definition\n\nLet $X$ be a real random variable on a probability space $(\\Omega, \\mathcal{F}, P)$.\nA probability density function of $X$ is a Borel measurable function $f \\colon \\mathbb{R} \\to [0, \\infty)$ such that $P(X \\in B) = \\int_B f(x) \\, dx$ for each Borel set $B \\subseteq \\mathbb{R}$.\nSo $X$ has a probability density function if and only if $X$ is a [[random variable with a density|0C21RZG5V2]].\n\n## Remarks\n\nEach density satisfies $\\int_{-\\infty}^{\\infty} f(x) \\, dx = 1$. Two densities of $X$ are equal except on a set of Lebesgue measure zero.\n$P(a \\le X \\le b) = \\int_a^b f(x) \\, dx$ for $a \\le b$.\nIf the [[cumulative distribution function|0MXMS3AE13]] $F_X$ is differentiable at $x$ and $f$ is continuous at $x$, then $F_X'(x) = f(x)$.\nA density can be larger than $1$. The uniform distribution on $[0, 1/2]$ has the density $2$ on this interval.\n\n## Explanations\n\n### L1 human\n\nThe density $f(x)$ is not a probability. It is a probability per unit of length near $x$.\nFor a small $h$, the probability that $X$ lies between $x$ and $x + h$ is about $f(x) \\, h$.\n",
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