Two-parameter Weibull Distribution
The probability density function of two parameter Weibull distribution is
$$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{\alpha}{\beta} \big(\frac{x}{\beta}\big)^{\alpha-1}e^{-\big(\frac{x}{\beta}\big)^\alpha}, & \hbox{$x>0$, $\alpha, \beta>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
- $\alpha$ is the shape parameter
- $\beta$ is the scale parameter
Mean of Two-parameter Weibull Distribution
The mean of Two-parameter Weibull distribution is
$E(X) = \beta \Gamma (\dfrac{1}{\alpha}+1)$.
Variance of Two-parameter Weibull Distribution
The variance of Two-parameter Weibull distribution is
$V(X) = \beta^2 \bigg(\Gamma (\dfrac{2}{\alpha}+1) -\bigg(\Gamma (\dfrac{1}{\alpha}+1) \bigg)^2\bigg)$
Distribution function of Weibull Distribution
The distribution function of Three-parameter Weibull distribution is
$F(x) = 1- e^{-\big(\frac{x-\mu}{\beta}\big)^\alpha}.$
Median of Weibull Distribution
The median of two parameter Weibull distribution is
