Pareto distribution

${\displaystyle x_{\mathrm {m} }>0}$ scale (real)
${\displaystyle \alpha >0}$ shape (real)

${\displaystyle x\in [x_{\mathrm {m} },\infty )}$

${\displaystyle {\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}}$

${\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }}$

${\displaystyle x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}}$

${\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}$

${\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}$

${\displaystyle x_{\mathrm {m} }}$

${\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 2\\{\dfrac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}$

${\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}$

${\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}$

${\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}$

does not exist

${\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}$

${\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}$

${\displaystyle {\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}}$^[1]

${\displaystyle L>0}$ location (real)
${\displaystyle H>L}$ location (real)

${\displaystyle \alpha >0}$

${\displaystyle L\leqslant x\leqslant H}$

${\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}$

${\displaystyle {\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}$

${\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -1}}\right)\cdot \left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),\alpha \neq 1}$

${\displaystyle {\frac {{H}{L}}{{H}-{L}}}\ln {\frac {H}{L}},\alpha =1}$

${\displaystyle L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}}$

${\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -2}}\right)\cdot \left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),\alpha \neq 2}$ ${\displaystyle {\frac {2{H}^{2}{L}^{2}}{{H}^{2}-{L}^{2}}}\ln {\frac {H}{L}},\alpha =2}$

${\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha (L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j}$