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Weibull Distribution

The Weibull distribution is given by

$\displaystyle P(x)$ $\textstyle =$ $\displaystyle \alpha\beta^{-\alpha} x^{\alpha-1}e^{-(x/\beta)^\alpha}$ (1)
$\displaystyle D(x)$ $\textstyle =$ $\displaystyle 1-e^{-(x/\beta)^\alpha}$ (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
$\displaystyle \mu$ $\textstyle =$ $\displaystyle \beta\Gamma(1+\alpha^{-1})$ (3)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \beta^2[\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]$ (4)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {2\Gamma^3(1+\alpha^{-1})-3\Gamma(1+\alpha^{-1})\Gamma(1+2\alpha^{-1})\over
{[}\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]^{3/2}}$  
  $\textstyle \phantom{=}$ $\displaystyle +{\Gamma(1+3\alpha^{-1})\over [\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]^{3/2}}$ (5)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {f(a)\over [\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})]^2},$ (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

$\displaystyle P(x)$ $\textstyle =$ $\displaystyle {\alpha\over\beta} x^{\alpha-1}e^{-x^\alpha/\beta}$ (8)
$\displaystyle D(x)$ $\textstyle =$ $\displaystyle 1-e^{-x^\alpha/\beta}$ (9)

(Mendenhall and Sincich 1995). The Mean and Variance for this form are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle \beta^{1/\alpha}\Gamma(1+\alpha^{-1})$ (10)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \beta^{2/\alpha}[\Gamma(1+2\alpha^{-1})-\Gamma^2(1+\alpha^{-1})].$ (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.



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© 1996-9 Eric W. Weisstein
1999-05-26