Probability distribution
Pareto Type I
Probability density function ![Pareto Type I probability density functions for various α](//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Probability_density_function_of_Pareto_distribution.svg/325px-Probability_density_function_of_Pareto_distribution.svg.png) Pareto Type I probability density functions for various with As the distribution approaches where is the Dirac delta function. |
Cumulative distribution function ![Pareto Type I cumulative distribution functions for various α](//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Cumulative_distribution_function_of_Pareto_distribution.svg/325px-Cumulative_distribution_function_of_Pareto_distribution.svg.png) Pareto Type I cumulative distribution functions for various with ![{\displaystyle x_{\mathrm {m} }=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96d6ad0977f3df766a42740e59fad636bdfc0012) |
Parameters |
scale (real)
shape (real) |
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Support |
![{\displaystyle x\in [x_{\mathrm {m} },\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1353d7fdc761ba14872a4b4225faa0583e3f1c95) |
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PDF |
![{\displaystyle {\frac {\alpha x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9540472e6411a94f070944a6e8cfc7978801ae36) |
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CDF |
![{\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f34628cd3a4b8221565660d2b1d4c41cabe8d6) |
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Quantile |
![{\displaystyle x_{\mathrm {m} }{(1-p)}^{-{\frac {1}{\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0df339f9fe83fd0de4aff256f9eec3b45c98db) |
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Mean |
![{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\dfrac {\alpha x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9948b5592808afb534fe7f331ba307f8ef4355) |
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Median |
![{\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1a9e02a1d60cf9cd611b13188b078509904bc7) |
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Mode |
![{\displaystyle x_{\mathrm {m} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddce2d6463b7cc6de9d022a861b92e450434fda) |
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Variance |
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf518cc574a4fc79a8ac942a6a75a9b19ab9a778) |
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Skewness |
![{\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/539b213d7dbe7ca0c2b864a03c02a25103556001) |
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Excess kurtosis |
![{\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02731e4b801e81407754b790466d88c32b7c96d2) |
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Entropy |
![{\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2581cbc359c3d0775610440ab0afeb1ef2f3819c) |
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MGF |
does not exist |
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CF |
![{\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a900a005ddef054f1c72031b63109a286a1c2f8) |
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Fisher information |
![{\displaystyle {\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e9daf9821878294e8d09162cad9f3623b5edfc) |
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Expected shortfall |
[1] |
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The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[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; the principle 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][7]