The Weibull distribution

The Weibull distribution is a common distribution used to describe the time-to-failure of aging components. Let $X$ be a Weibull distributed stochastic variable. The probability density function of $X$ is given gitt by:

$ \color{blue}{f_{X}(x)=\alpha \lambda (\lambda x)^{\alpha -1}e^{-(\lambda x)^{\alpha}}} $

The cummulative distribution function of X is given by:

$ \color{blue}{F_{X}(x) = 1- e^{-(\lambda x) ^{\alpha}}} $

where α is a shape (aging) parameter and λ is an intensity parameter.

Expectation and variance is given by:

$ \color{blue}{\mathrm{E}(X)=\frac{1}{\lambda } \Gamma \left( \frac{1}{\alpha }+1 \right) } $

$ \color{blue}{\mathrm{Var}(X) = \frac{1}{\lambda ^2}\left( {\Gamma \left( \frac{2 }{\alpha }+1)-\Gamma ^2(\frac{1 }{ \alpha }+1 \right)} \right)} $

where Γ(⋅) is the gamma function. MS Excel provides it by: =Gamma(x). Below a sketch of the Weibull distribution with α = 2 and λ = 0.001 is given.

The failure rate function

The Weibull distribution has a failure rate function on the following form: $ \color{blue}{z(t)=\alpha \lambda^{\alpha} t^{\alpha -1}}$, where we have introduced t as the variable because we deal with time.

The failure rate function is:

Increasing for α > 1 (IFR=increasing failure rate)
Decreasing for α < 1 (DFR=decreasing failure rate)
Constant for α = 1

For the Weibull distribution the coefficient of variance is given by:

$ \color{blue}{{\mathrm{cv}}= \sqrt{ \frac {\Gamma(1+2/\alpha )}{ \Gamma^2(1+1/\alpha)} -1 }} $

and as we observe it only depends on the shape (aging) coefficient.