Editing Weibull distribution (section)

====Second alternative====
A second alternative parameterization can also be found.<ref>{{Cite book|last1=Kalbfleisch|first1=J. D.|title=The statistical analysis of failure time data|last2=Prentice|first2=R. L.|publisher=J. Wiley|year=2002|isbn=978-0-471-36357-6|edition=2nd|location=Hoboken, N.J.|oclc=50124320}}</ref><ref>{{Cite web|last=Therneau|first=T.|date=2020|others=R package version 3.1.|title=A Package for Survival Analysis in R.|url=https://CRAN.R-project.org/package=survival}}</ref> The shape parameter ''k'' is the same as in the standard case, while the scale parameter ''&lambda;'' is replaced with a rate parameter ''&beta;'' = 1/''&lambda;''. Then, for ''x'' ≥ 0, the probability density function is
:<math>f(x;k,\beta) = \beta k({\beta x})^{k-1} e^{-(\beta x)^k}</math>
the cumulative distribution function is 
:<math>F(x;k,\beta) = 1 - e^{-(\beta x)^k},</math>
the quantile function is 
:<math>Q(p;k,\beta) = \frac{1}{\beta}(-\ln(1-p))^\frac{1}{k},</math>
and the hazard function is
:<math>h(x;k,\beta) = \beta k({\beta x})^{k-1}.</math>

In all three parameterizations, the hazard is decreasing for k < 1, increasing for k > 1 and constant for k = 1, in which case the Weibull distribution reduces to an exponential distribution.