Weibull

The Weibull distribution with scale parameterb > 0 and shape parameter a > 0 is defined by the integral

$\displaystyle \func$WeibullDist$\displaystyle \left(\vphantom{ x;a,b}\right.$x;a, b$\displaystyle \left.\vphantom{ x;a,b}\right)$ = ab-a$\displaystyle \int_{{0}}^{{x}}$ua-1e-uab-adu = 1 - e-xab-a

of the density function

$\displaystyle \limfunc$WeibullDen(u;a, b) = ab-aua-1e-uab-a

for x≥ 0, and is 0 otherwise.

The inverse Weibull distribution function

$\displaystyle \limfunc$WeibullInv$\displaystyle \left(\vphantom{ \alpha ;a,b}\right.$α;a, b$\displaystyle \left.\vphantom{ \alpha ;a,b}\right)$ = b$\displaystyle \left(\vphantom{ \ln \frac{1}{1-\alpha
}}\right.$ln$\displaystyle {\frac{{1}}{{1-\alpha
}}}$$\displaystyle \left.\vphantom{ \ln \frac{1}{1-\alpha
}}\right)^{{\frac{1}{a}}}_{}$

is the value of x for which the  integral has the value   α, as illustrated by the following:

$\blacktriangleright$ Evaluate

$\limfunc$WeibullInv$\left(\vphantom{ .73;.5,.3}\right.$.73;.5,.3$\left.\vphantom{ .73;.5,.3}\right)$ = .3$\left(\vphantom{ \ln \frac{1}{1-.73}%
}\right.$ln${\frac{{1}}{{1-.73}%
}}$$\left.\vphantom{ \ln \frac{1}{1-.73}%
}\right)^{{\frac{1}{.5}}}_{}$ =  .51431 

$\func$WeibullDist$\left(\vphantom{ .51431;.5,.3}\right.$.51431;.5,.3$\left.\vphantom{ .51431;.5,.3}\right)$ = 1 - e-.51431.5.3-.5 =  .73

The following plots show cumulative distribution functions $\func$WeibullDist$\left(\vphantom{ x;a,b}\right.$x;a, b$\left.\vphantom{ x;a,b}\right)$ and density functions $\limfunc$WeibullDen(x;a, b), for the parameters (a, b) = (.5, 1),(1, 1), (3,.5),(3, 1) and 0≤x≤3.

itbpFU2.1577in1.7573in0inWeibull density functions itbpFU2.1577in1.7573in0inWeibull distribution functions