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Pareto Distribution

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ParetoDistribution

The distribution with probability density function and distribution function

P(x)=(ab^a)/(x^(a+1))
(1)
D(x)=1-(b/x)^a
(2)

defined over the interval x>=b.

It is implemented in the Wolfram Language as ParetoDistribution[k, alpha].

The nth raw moment is

mu_n^'=(ab^n)/(a-n)
(3)

for a>n, giving the first few as

mu_1^'=(ab)/(a-1)
(4)
mu_2^'=(ab^2)/(a-2)
(5)
mu_3^'=(ab^3)/(a-3)
(6)
mu_4^'=(ab^4)/(a-4).
(7)

The nth central moment is

mu_n=ab^nGamma(a-n)_2F^~_1(a-n,-n;1+a-n;a/(a-1))
(8)
=(1-a)^(a-n)(-a)^(n-a)ab^nB(a/(a-1);a-n,n+1),
(9)

for a>n and where Gamma(z) is a gamma function, _2F^~_1(a,b;c;z) is a regularized hypergeometric function, and B(z;a,b) is a beta function, giving the first few as

mu_2=(ab^2)/((a-1)^2(a-2))
(10)
mu_3=(2a(a+1)b^3)/((a-1)^3(a-2)(a-3))
(11)
mu_4=(3a(3a^3+a+2)b^4)/((a-1)^4(a-2)(a-3)(a-4)).
(12)

The mean, variance, skewness, and kurtosis excess are therefore

mu=(ab)/(a-1)
(13)
sigma^2=(ab^2)/((a-1)^2(a-2))
(14)
gamma_1=sqrt((a-2)/a)(2(a+1))/(a-3)
(15)
gamma_2=(6(a^3+a^2-6a-2))/(a(a-3)(a-4)).
(16)

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References

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 252, 1993.

Referenced on Wolfram|Alpha

Pareto Distribution

Cite this as:

Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParetoDistribution.html

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