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  "body": "\n## Example\n\nLet $x_m \u003e 0$ and $\\alpha \u003e 0$.\nA random variable $X$ has the *Pareto distribution* with scale $x_m$ and shape $\\alpha$ if\n$$P(X \u003e x) = \\left(\\frac{x_m}{x}\\right)^{\\alpha} \\quad \\text{for } x \\ge x_m, \\qquad P(X \u003e x) = 1 \\quad \\text{for } x \u003c x_m.$$\nThis is a valid law: the [[distribution function|0H8H7ZTHW2]] $1 - (x_m/x)^\\alpha$ is continuous and increasing from $0$ at $x_m$ to the limit $1$ as $x \\to \\infty$.\nIts [[density|0ZY9DX8FBF]] is $f(x) = \\alpha x_m^{\\alpha} / x^{\\alpha + 1}$ for $x \\ge x_m$ and $0$ otherwise.\nFor $k \u003e 0$, $E[X^k] = \\int_{x_m}^\\infty \\alpha x_m^\\alpha x^{k - \\alpha - 1} \\, dx$ is finite if and only if $k \u003c \\alpha$, and then equals $\\frac{\\alpha x_m^k}{\\alpha - k}$.\nSo the mean is $\\frac{\\alpha x_m}{\\alpha - 1}$ for $\\alpha \u003e 1$ and infinite for $\\alpha \\le 1$, and the variance is $\\frac{\\alpha x_m^2}{(\\alpha - 1)^2 (\\alpha - 2)}$ for $\\alpha \u003e 2$.\n\n## Remarks\n\nThe variable $\\ln(X / x_m)$ is [[exponential|0E99F51GVB]] with rate $\\alpha$, since $P(\\ln(X/x_m) \u003e s) = P(X \u003e x_m e^s) = e^{-\\alpha s}$ for $s \\ge 0$.\nThe law is scale free in the following sense: for $y \\ge x_m$, given $X \u003e y$ the variable $X$ is Pareto with scale $y$ and the same shape $\\alpha$.\nIt is the basic example of a heavy-tailed law, with a tail that decays like a power of $x$; only the [[moments|01XAHPADBA]] of order less than $\\alpha$ are finite.\n\n## Explanations\n\n### L1 human\n\nA Pareto variable is at least some minimum value, and the chance that it exceeds $x$ falls like a power $x^{-\\alpha}$.\nDoubling the threshold always divides the chance of exceeding it by the same factor $2^\\alpha$, at every scale.\nFor small $\\alpha$, rare huge values dominate, and the mean can even be infinite.\nVilfredo Pareto used such laws to describe the distribution of wealth, and they also model city sizes and insurance losses.\n",
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