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Pearson Type III Distribution

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A skewed distribution which is similar to the binomial distribution when p!=q (Abramowitz and Stegun 1972, p. 930).

y=k(t+A)^(A^2-1)e^(-At),
(1)

for t in [0,infty) where

A=2/gamma
(2)
K=(A^(A^2)e^(-A^2))/(Gamma(A^2)),
(3)

Gamma(z) is the gamma function, and T is a standardized variate. Another form is

P(x)=1/(betaGamma(p))((x-alpha)/beta)^(p-1)exp(-(x-alpha)/beta).
(4)

For this distribution, the characteristic function is

phi(t)=e^(ialphat)(1-ibetat)^(-p),
(5)

and the mean, variance, skewness, and kurtosis excess are

mu=alpha+pbeta
(6)
sigma^2=pbeta^2
(7)
gamma_1=2/(sqrt(p))
(8)
gamma_2=6/p.
(9)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

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Pearson Type III Distribution

Cite this as:

Weisstein, Eric W. "Pearson Type III Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PearsonTypeIIIDistribution.html

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