Template:Mixed weibull equations: Difference between revisions
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::<math>{{R}_{1,...,S}}( | ::<math>{{R}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}}</math> | ||
and: | |||
::<math>{{f}_{1,...,S}}( | ::<math>{{f}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{t}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}}</math> | ||
where <math>S=2</math> , <math>S=3</math> , and <math>S=4</math> for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters. | where <math>S=2</math> , <math>S=3</math> , and <math>S=4</math> for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters. | ||
Revision as of 18:01, 14 February 2012
The Mixed Weibull Equations
Depending on the number of subpopulations chosen, Weibull++ uses the following equations for the reliability and probability density functions:
- [math]\displaystyle{ {{R}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}} }[/math]
and:
- [math]\displaystyle{ {{f}_{1,...,S}}(t)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{t}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{t}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}} }[/math]
where [math]\displaystyle{ S=2 }[/math] , [math]\displaystyle{ S=3 }[/math] , and [math]\displaystyle{ S=4 }[/math] for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.