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Pareto Type I

Probability density function Pareto Type I probability density functions for various $\alpha$ with $x_{\mathrm {m} }=1.$ As $\alpha \rightarrow \infty ,$ the distribution approaches $\delta (x-x_{\mathrm {m} }),$ where $\delta$ is the Dirac delta function.
Cumulative distribution function Pareto Type I cumulative distribution functions for various $\alpha$ with $x_{\mathrm {m} }=1.$
Parameters	$x_{\mathrm {m} }>0$ scale (real) $\alpha >0$ shape (real)
Support	$x\in [x_{\mathrm {m} },\infty )$
PDF	${\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}$
CDF	$1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }$
Quantile	$x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}$
Mean	${\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}$
Median	$x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}$
Mode	$x_{\mathrm {m} }$
Variance	${\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}$
Skewness	${\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3$
Ex. kurtosis	${\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4$
Entropy	$\log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)$
MGF	does not exist
CF	$\alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)$
Fisher information	$\mathcal{I}(x_{\mathrm{m}},\mathrm{\alpha <!--\; \alpha \; -->})=[\begin{array}{cc}\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{x_{\mathrm{m}}^2}& \end{array}$