Weibull Distribution

The Weibull distribution is given by

$$P(x) = \alpha\beta^{-\alpha} x^{\alpha-1}e^{-(x/\beta)^\alpha}$$ (1)

$$D(x) = 1-e^{-(x/\beta)^\alpha}$$ (2)

for $x\in [0,\infty)$ (Mathematica ${}^{\scriptstyle\circledRsymbol}$ Statistics`ContinuousDistributions`WeibullDistribution[a,b], Wolfram Research, Champaign, IL). The Mean, Variance, Skewness, and Kurtosis of this distribution are

$$\mu = \beta\Gamma(1+\alpha^{-1})$$ (3)

$$\sigma^2 = \beta^2[\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]$$ (4)

$$\gamma_1 = {2\Gamma^3(1+\alpha^{-1})-3\Gamma(1+\alpha^{-1})\Gamma(1+2\alpha^{-1})\over {[}\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]^{3/2}}$$

$$\phantom{=} +{\Gamma(1+3\alpha^{-1})\over [\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]^{3/2}}$$ (5)

$$\gamma_2 = {f(a)\over [\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]^2},$$ (6)

where $\Gamma(z)$ is the Gamma Function and

$f(a)\equiv -6\Gamma^4(1+\alpha^{-1})+12\Gamma^2(1+\alpha^{-1})\Gamma(1+2\alpha^{-1})$
$-3\Gamma^2(1+2\alpha^{-1})-4\Gamma(1+\alpha^{-1})\Gamma(1+3\alpha^{-1})+\Gamma(1+4\alpha^{-1}).\quad$	(7)

A slightly different form of the distribution is

$$P(x) = {\alpha\over\beta} x^{\alpha-1}e^{-x^\alpha/\beta}$$ (8)

$$D(x) = 1-e^{-x^\alpha/\beta}$$ (9)

(Mendenhall and Sincich 1995). The Mean and Variance for this form are

$$\mu = \beta^{1/\alpha}\Gamma(1+\alpha^{-1})$$ (10)

$$\sigma^2 = \beta^{2/\alpha}[\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})].$$ (11)

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 Fisher-Tippett Distribution

References

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.