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Statistics[Distributions]

 Weibull
 Weibull distribution

 Calling Sequence Weibull(b, c) WeibullDistribution(b, c)

Parameters

 b - scale parameter c - shape parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{c{t}^{c-1}{ⅇ}^{-{\left(\frac{t}{b}\right)}^{c}}}{{b}^{c}}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Weibull}\left(b,c\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{c}{}{{u}}^{{c}{-}{1}}{}{{ⅇ}}^{{-}{\left(\frac{{u}}{{b}}\right)}^{{c}}}}{{{b}}^{{c}}}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{c}{}{{0.5}}^{{-}{1.}{+}{c}}{}{{ⅇ}}^{{-}{1.}{}{\left(\frac{{0.5}}{{b}}\right)}^{{c}}}}{{{b}}^{{c}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${b}{}{\mathrm{\Gamma }}{}\left(\frac{{1}{+}{c}}{{c}}\right)$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{b}}^{{2}}{}\left({\mathrm{\Gamma }}{}\left(\frac{{c}{+}{2}}{{c}}\right){-}{{\mathrm{\Gamma }}{}\left(\frac{{1}{+}{c}}{{c}}\right)}^{{2}}\right)$ (4)

References

 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.

 See Also