The Weibull distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The Weibull variate is related to the standard Weibull variate by Weibull(b,c) ~ b*Weibull(1,c).

The Weibull variate with scale parameter b and shape parameter 1 is equivalent to the Exponential variate with scale parameter b:  Weibull(b,1) ~ Exponential(b).

The Weibull variate with scale parameter b and shape parameter 2 is equivalent to the Rayleigh variate:  Weibull(b,2) ~ Rayleigh(b).

Note that the Weibull command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Weibull}\left(b\,c\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{c{u}^{c-1}{\&ExponentialE;}^{-{\left(\frac{u}{b}\right)}^c}}{b^c}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{c{0.5}^{-1.+c}{\&ExponentialE;}^{-1.{\left(\frac{0.5}{b}\right)}^c}}{b^c}$

$\mathrm{Mean}\left(X\right)$

$b\mathrm{\Gamma }\left(\frac{1+c}{c}\right)$

$\mathrm{Variance}\left(X\right)$

$b^2\left(\mathrm{\Gamma }\left(\frac{c+2}{c}\right)-{\mathrm{\Gamma }\left(\frac{1+c}{c}\right)}^2\right)$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.