Weibull distribution

Standard parameterization

The probability density function of a Weibull random variable is:

\[{\displaystyle f(x;\lambda ,k)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}&x\geq 0,\\0&x<0,\end{cases}}}\]

where \(k > 0\) is the shape parameter and \(\lambda > 0\) is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function.

The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (\(k = 1\)) and the Rayleigh distribution (\(k = 2\)) and \[{\displaystyle \lambda ={\sqrt {2}}\sigma }\].

Density Plot of Random Covariates

• Shape Parameter set to 2,3,5 and 10.

Weibull = data.frame(
  k2 <- rweibull(1000,shape=2),
  k3 <- rweibull(1000,shape=3),
  k5 <- rweibull(1000,shape=5),
  k10 <- rweibull(1000,shape=10)
)

## Warning: package 'ggplot2' was built under R version 4.0.5