Template:Mixed weibull equations

The Mixed Weibull Equations
Depending on the number of subpopulations chosen, Weibull++ uses the following equations for the reliability and probability density functions:


 * $${{R}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{N}{{e}^{-{{\left( \tfrac{t}{{{\eta }_{i}}} \right)}^}}}$$

and:


 * $${{f}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{N{{\eta }_{i}}}{{\left( \frac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{t}{{{\eta }_{i}}})}^}}}$$

where $$S=2$$,  $$S=3$$ , and  $$S=4$$  for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.