Template:Weibull Probability Density Function

The Three-Parameter Weibull Distribution
The three-parameter Weibull pdf is given by:


 * $$ f(T)={ \frac{\beta }{\eta }}\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }} $$


 * where,


 * $$ f(T)\geq 0,\text{ }T\geq 0\text{ or }\gamma, $$


 * $$\beta>0\ \,\!$$,


 * $$ \eta > 0 \,\!$$,


 * $$ -\infty < \gamma < +\infty \,\!$$


 * and,


 * $$ \eta= \,\!$$ scale parameter, or characteristic life
 * $$ \beta= \,\!$$ shape parameter (or slope),
 * $$ \gamma= \,\!$$ location parameter (or failure free life).

The Two-Parameter Weibull Distribution
The two-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 One-Parameter Weibull Distribution
The one-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 one-parameter Weibull, we assume that the shape parameter $$\beta \,\!$$ is known a priori from past experience on identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.