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Weibull Distribution

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The Weibull distribution is given by

P(x)=alphabeta^(-alpha)x^(alpha-1)e^(-(x/beta)^alpha)
(1)
D(x)=1-e^(-(x/beta)^alpha)
(2)

for x in [0,infty), and is implemented in the Wolfram Language as WeibullDistribution[alpha, beta]. The raw moments of the distribution are

mu_1^'=betaGamma(1+alpha^(-1))
(3)
mu_2^'=beta^2Gamma(1+2alpha^(-1))
(4)
mu_3^'=beta^3Gamma(1+3alpha^(-1))
(5)
mu_4^'=beta^4Gamma(1+4alpha^(-1)),
(6)

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

mu=betaGamma(1+alpha^(-1))
(7)
sigma^2=beta^2[Gamma(1+2alpha^(-1))-Gamma^2(1+alpha^(-1))]
(8)
gamma_1=(2Gamma^3(1+alpha^(-1))-3Gamma(1+alpha^(-1))Gamma(1+2alpha^(-1)))/([Gamma(1+2alpha^(-1))-Gamma^2(1+alpha^(-1))]^(3/2))+(Gamma(1+3alpha^(-1)))/([Gamma(1+2alpha^(-1))-Gamma^2(1+alpha^(-1))]^(3/2))
(9)
gamma_2=(f(alpha))/([Gamma(1+2alpha^(-1))-Gamma^2(1+alpha^(-1))]^2),
(10)

where Gamma(z) is the gamma function and

f(alpha)=-6Gamma^4(1+alpha^(-1))+12Gamma^2(1+alpha^(-1))Gamma(1+2alpha^(-1))-3Gamma^2(1+2alpha^(-1))-4Gamma(1+alpha^(-1))Gamma(1+3alpha^(-1))+Gamma(1+4alpha^(-1)).
(11)

A slightly different form of the distribution is defined by

P(x)=alpha/betax^(alpha-1)e^(-x^alpha/beta)
(12)
D(x)=1-e^(-x^alpha/beta)
(13)

(Mendenhall and Sincich 1995). This has raw moments

mu_1^'=beta^(1/alpha)Gamma(1+alpha^(-1))
(14)
mu_2^'=beta^(2/alpha)Gamma(1+2alpha^(-1))
(15)
mu_3^'=beta^(3/alpha)Gamma(1+3alpha^(-1))
(16)
mu_4^'=beta^(4/alpha)Gamma(1+4alpha^(-1)),
(17)

so the mean and variance for this form are

mu=beta^(1/alpha)Gamma(1+alpha^(-1))
(18)
sigma^2=beta^(2/alpha)[Gamma(1+2alpha^(-1))-Gamma^2(1+alpha^(-1))].
(19)

The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in analysis of systems involving a "weakest link."

See also

Extreme Value Distribution, Gumbel Distribution

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References

Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: Wiley, 1995.Kobayashi, A. (Ed.). Handbook on Experimental Mechanics. New York: VCH/SEM, 1993.Mendenhall, W. and Sincich, T. Statistics for Engineering and the Sciences, 4th ed. Englewood Cliffs, NJ: Prentice Hall, 1995.Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 119, 1992.

Referenced on Wolfram|Alpha

Weibull Distribution

Cite this as:

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

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