= Pareto distribution
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{title2=$\operatorname{Pareto}(\alpha,\beta)$}
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The <Pareto distribution> describes a positive <random variable> whose <tail probability> falls as a <power law>. In the shape–minimum convention used here, $X\sim\operatorname{Pareto}(\alpha,\beta)$ has <shape parameter> $\alpha>0$ and minimum $\beta>0$, with <probability support> $[\beta,\infty)$. Its <probability density function> is
$$
f(x)=\begin{cases}\alpha\beta^\alpha x^{-\alpha-1},&x\geq\beta,\\0,&x<\beta.\end{cases}
$$
The minimum $\beta$ has the same units as $X$; the <shape parameter> $\alpha$ is dimensionless. At fixed $\alpha$, changing $\beta$ gives a <scale family>, since $X/\beta\sim\operatorname{Pareto}(\alpha,1)$. This convention is also used in <uniform-Pareto conjugacy>.<NIST Dataplot Reference Manual/Pareto Probability Density Function>[NIST Dataplot Reference Manual; Description: standard Pareto density and lower bound; source gamma,a correspond to alpha,beta here]{ref}
The <survival function> and <cumulative distribution function> are
$$
\overline F(x)=\mathbb P(X>x)=\begin{cases}1,&x<\beta,\\(\beta/x)^\alpha,&x\geq\beta,\end{cases}
\qquad F(x)=1-\overline F(x).
$$
For $0<p<1$, solving $F(x)=p$ gives the <quantile function>
$$
Q(p)=\beta(1-p)^{-1/\alpha}.
$$
Its right-hand limit at $p=0$ is $\beta$; it tends to infinity as $p\uparrow1$.<Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist/The Pareto Distribution>[Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist; The General Pareto Distribution, Distribution Functions: density, CDF and quantiles; source a,b correspond to alpha,beta here]{ref} There is no <probability mass> at $\beta$: changing a <probability density function>'s value at that single point changes no <probabilities>.
Choose $\alpha=2$ and $\beta=10$ as an original theoretical example. Then $\mathbb P(X>20)=1/4$, $\mathbb P(X>100)=1/100$, the <median> is $10\sqrt2\simeq14.142$, and the 0.99 <quantile> is $Q(0.99)=100$. These chosen values and all figures below describe mathematical <probability distributions>, not actual wealth observations.
\Image[/pareto-survival-loglog.png]
{title=Pareto survival curves: smaller alpha gives a slower power-law tail}
On this <log-log plot>, write $r=x/\beta$ to obtain $\log\overline F(\beta r)=-\alpha\log r$. The <slope> is $-\alpha$, whereas the <probability density function> has power exponent $\alpha+1$. Multiplying a threshold by ten divides its <tail probability> by $10^\alpha$. A larger $\alpha$ therefore concentrates the <Pareto distribution> nearer $\beta$; a smaller $\alpha$ makes extreme values more plausible. This exact scaling is useful when modelling large positive losses or sizes over several orders of magnitude.
The <Pareto distribution> is a <heavy-tailed distribution>: for every $t>0$, the <exponential moment> $\mathbb E[e^{tX}]$ diverges because $e^{tx}x^{-\alpha-1}$ eventually grows without bound. In contrast, the <Laplace distribution> has exponentially decreasing tails and some finite positive <exponential moments>. The <Student's t-distribution> also has power tails, but on both sides of the <real line>, with a smooth central region rather than a positive minimum.
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