Pareto Distribution
ParetoDistribution
The distribution with probability density function and distribution function
P(x) = (ab^a)/(x^(a+1))
(1)
D(x) = [画像: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
| [画像: mu_n^'=(ab^n)/(a-n) ] |
(3)
|
for a>n, giving the first few as
mu_1^' = [画像:(ab)/(a-1)]
(4)
mu_2^' = [画像:(ab^2)/(a-2)]
(5)
mu_3^' = [画像:(ab^3)/(a-3)]
(6)
mu_4^' = [画像:(ab^4)/(a-4).]
(7)
The nth central moment is
mu_n = [画像:ab^nGamma(a-n)_2F^~_1(a-n,-n;1+a-n;a/(a-1))]
(8)
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 = [画像:(ab^2)/((a-1)^2(a-2))]
(10)
mu_3 = [画像:(2a(a+1)b^3)/((a-1)^3(a-2)(a-3))]
(11)
mu_4 = [画像:(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 = [画像:(ab)/(a-1)]
(13)
sigma^2 = [画像:(ab^2)/((a-1)^2(a-2))]
(14)
gamma_1 = [画像:sqrt((a-2)/a)(2(a+1))/(a-3)]
(15)
gamma_2 = [画像:(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 DistributionCite this as:
Weisstein, Eric W. "Pareto Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ParetoDistribution.html