xbeta
Weibull distribution

The Weibull distribution (Weibull, 1951) has cdf

and pdf

Here, $\lambda >0$ is a scale parameter and $\alpha >0$ is a shape parameter.

Special Cases

For certain values of $\alpha$, the Weibull distribution reduces to other common distributions: The Weibull distribution with $\alpha =1$ reduces to the Exponential distribution with rate parameter $\lambda$.

• When $\alpha =1$, it is equivalent to the Exponential distribution with rate parameter $\lambda$.
• When $\alpha =2$, it is equivalent to the Rayleigh distribution? with variance ${\lambda }^2/2$.
• When $\alpha =3.4$, it is approximately the univariate Normal distribution.
• As $\alpha \to \mathrm{\infty }$, it converges to the Dirac delta function?.

References

Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech.-Trans. ASME 18, 293–297.

External Links

• A Statistical Distribution Function of Wide Applicability (the original 1951 article)
• Wikipedia:Weibull distribution