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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers.\nThe continuous uniform distribution on the [[interval|0TK8Y3J5RW]] $[a, b]$ is the distribution on $\\mathbb{R}$ with the [[probability density function|04YG690N5V]]\n$$f(x) = \\begin{cases} \\dfrac{1}{b - a} \u0026 \\text{if } a \\le x \\le b, \\\\ 0 \u0026 \\text{otherwise.} \\end{cases}$$\nOne writes $X \\sim \\operatorname{U}(a, b)$.\n\n## Remarks\n\nFor $a \\le c \\le d \\le b$, $P(c \\le X \\le d) = (d - c)/(b - a)$.\nThe cumulative distribution function is $F(x) = (x - a)/(b - a)$ for $a \\le x \\le b$, with $F(x) = 0$ for $x \u003c a$ and $F(x) = 1$ for $x \u003e b$.\nThe mean is $(a + b)/2$ and the variance is $(b - a)^2 / 12$.\nIf $U \\sim \\operatorname{U}(0, 1)$ and $F$ is a cumulative distribution function with [[quantile function|0B3Q3XDM5X]] $Q$, then $Q(U)$ has the cumulative distribution function $F$.\n\n## Explanations\n\n### L1 human\n\nThe uniform distribution is \"a point chosen at random in an interval, with no preference\".\nFor $\\operatorname{U}(0, 10)$, the chance to land between $2$ and $5$ is $3/10$, the length of the piece over the length of the interval.\n",
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