Pareto distribution information

Pareto Type I
	Probability density function; 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 with
Parameters	scale (real); shape (real)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF	does not exist
CF
Fisher information
Expected shortfall

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]

^ ^a ^b Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Retrieved 2023-02-27.
^ Amoroso, Luigi (1938). "VILFREDO PARETO". Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest. 6.
^ Pareto, Vilfredo (1898). "Cours d'economie politique". Journal of Political Economy. 6. doi:10.1086/250536.
^ Cite error: The named reference :1 was invoked but never defined (see the help page).
^ VAN MONTFORT, M.A.J. (1986). "The Generalized Pareto distribution applied to rainfall depths". Hydrological Sciences Journal. 31 (2): 151–162. Bibcode:1986HydSJ..31..151V. doi:10.1080/02626668609491037.
^ Oancea, Bogdan (2017). "Income inequality in Romania: The exponential-Pareto distribution". Physica A: Statistical Mechanics and Its Applications. 469: 486–498. Bibcode:2017PhyA..469..486O. doi:10.1016/j.physa.2016.11.094.
^ Morella, Matteo. "Pareto Distribution". academia.edu.

[norton-1] Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Retrieved 2023-02-27.

[2] Amoroso, Luigi (1938). "VILFREDO PARETO". Econometrica (Pre-1986); Jan 1938; 6, 1; ProQuest. 6.

[3] Pareto, Vilfredo (1898). "Cours d'economie politique". Journal of Political Economy. 6. doi:10.1086/250536.

[:1-4] Cite error: The named reference :1 was invoked but never defined (see the help page).

[5] VAN MONTFORT, M.A.J. (1986). "The Generalized Pareto distribution applied to rainfall depths". Hydrological Sciences Journal. 31 (2): 151–162. Bibcode:1986HydSJ..31..151V. doi:10.1080/02626668609491037.

[6] Oancea, Bogdan (2017). "Income inequality in Romania: The exponential-Pareto distribution". Physica A: Statistical Mechanics and Its Applications. 469: 486–498. Bibcode:2017PhyA..469..486O. doi:10.1016/j.physa.2016.11.094.

[7] Morella, Matteo. "Pareto Distribution". academia.edu.

Pareto distribution information

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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	${\frac {2(1+\alpha )}{\alpha -3}}{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3$
Excess 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	${\mathcal {I}}(x_{\mathrm {m} },\alpha )={\begin{bmatrix}{\dfrac {\alpha ^{2}}{x_{\mathrm {m} }^{2}}}&0\\0&{\dfrac {1}{\alpha ^{2}}}\end{bmatrix}}$
Expected shortfall	${\frac {x_{m}\alpha }{(1-p)^{\frac {1}{\alpha }}(\alpha -1)}}$ ^[1]