The Weibull Distribution

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, $$ {\beta} \,\!$$. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++ software.

The 2-Parameter Weibull
The 2-parameter Weibull pdf is obtained by setting $$ \gamma=0 \,\!$$, and is given by:


 * $$ f(t)={ \frac{\beta }{\eta }}\left( {\frac{t}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{t}{\eta }}\right) ^{\beta }} \,\!$$

The 1-Parameter Weibull
The 1-parameter Weibull pdf is obtained by again setting $$\gamma=0 \,\!$$ and assuming $$\beta=C=Constant \,\!$$ assumed value or:


 * $$ f(t)={ \frac{C}{\eta }}\left( {\frac{t}{\eta }}\right) ^{C-1}e^{-\left( {\frac{t}{ \eta }}\right) ^{C}} \,\!$$

where the only unknown parameter is the scale parameter, $$\eta\,\!$$.

Note that in the formulation of the 1-parameter Weibull, we assume that the shape parameter $$\beta \,\!$$ is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.