# Pareto distribution

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[1] (Italian:  pə-RAY-toh),[2] is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena. Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.[3][4] The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value (α) of log45  1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena[5] and human activities.[6]

Parameters Probability density functionPareto Type I probability density functions for various ${\displaystyle \alpha }$ with ${\displaystyle x_{\mathrm {m} }=1.}$ As ${\displaystyle \alpha \rightarrow \infty ,}$ the distribution approaches ${\displaystyle \delta (x-x_{\mathrm {m} }),}$ where ${\displaystyle \delta }$ is the Dirac delta function. Cumulative distribution functionPareto Type I cumulative distribution functions for various ${\displaystyle \alpha }$ with ${\displaystyle x_{\mathrm {m} }=1.}$ ${\displaystyle x_{\mathrm {m} }>0}$ 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 ${\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}$ ${\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha }{x_{\mathrm {m} }^{2}}}&-{\dfrac {1}{x_{\mathrm {m} }}}\\-{\dfrac {1}{x_{\mathrm {m} }}}&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}$ Right: ${\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}$