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===The Mixed Weibull Distribution=== | === The Mixed Weibull Distribution === | ||
The mixed Weibull distribution is commonly used for modeling the behavior of components or systems exhibiting multiple failure modes (mixed populations). It gives the global picture of the life of a product by mixing different Weibull distributions for different stages of the product’s life and is defined by: | |||
<br> | The mixed Weibull distribution is commonly used for modeling the behavior of components or systems exhibiting multiple failure modes (mixed populations). It gives the global picture of the life of a product by mixing different Weibull distributions for different stages of the product’s life and is defined by: <br> | ||
::<math>f_{S}(t)=\sum_{i=1}^{S}p_{i}\frac{\beta_{i}}{\eta_{i}}(\frac{t}{\eta_{i}})^{\beta_{i}-1}e^{-(\frac{t}{\eta_{i}})^{\beta_{i}}} | ::<math>f_{S}(t)=\sum_{i=1}^{S}p_{i}\frac{\beta_{i}}{\eta_{i}}(\frac{t}{\eta_{i}})^{\beta_{i}-1}e^{-(\frac{t}{\eta_{i}})^{\beta_{i}}} | ||
</math> | </math> | ||
where the value of | where the value of <span class="texhtml">''S''</span> is equal to the number of subpopulations. Note that this results in a total of <math>(3\cdot S-1)</math> parameters. In other words, each population has a portion or mixing weight for the <span class="texhtml">''i''<sup>''t''''h'''''h''</sup></span> population, a <span class="texhtml">β<sub>''i''</sub></span> , or shape parameter for the <span class="texhtml">''i''<sup>''t'''</sup></span> population and or scale parameter <span class="texhtml">η<sub>''i''</sub></span> for <span class="texhtml">''i''<sup>''t''''h'''</sup></span> population. Note that the parameters are reduced to <math>(3\cdot S-1)</math>, given the fact that the following condition can also be used: <br> | ||
<br> | |||
::<math>\sum_{i=1}^{s}p_{i}=1</math> | ::<math>\sum_{i=1}^{s}p_{i}=1</math> | ||
The mixed Weibull distribution and its characteristics are presented in | The mixed Weibull distribution and its characteristics are presented in detail in the chapter [[The Mixed Weibull Distribution]]. | ||
Revision as of 16:18, 12 March 2012
The Mixed Weibull Distribution
The mixed Weibull distribution is commonly used for modeling the behavior of components or systems exhibiting multiple failure modes (mixed populations). It gives the global picture of the life of a product by mixing different Weibull distributions for different stages of the product’s life and is defined by:
- [math]\displaystyle{ f_{S}(t)=\sum_{i=1}^{S}p_{i}\frac{\beta_{i}}{\eta_{i}}(\frac{t}{\eta_{i}})^{\beta_{i}-1}e^{-(\frac{t}{\eta_{i}})^{\beta_{i}}} }[/math]
where the value of S is equal to the number of subpopulations. Note that this results in a total of [math]\displaystyle{ (3\cdot S-1) }[/math] parameters. In other words, each population has a portion or mixing weight for the it''hh population, a βi , or shape parameter for the it population and or scale parameter ηi for it'h population. Note that the parameters are reduced to [math]\displaystyle{ (3\cdot S-1) }[/math], given the fact that the following condition can also be used:
- [math]\displaystyle{ \sum_{i=1}^{s}p_{i}=1 }[/math]
The mixed Weibull distribution and its characteristics are presented in detail in the chapter The Mixed Weibull Distribution.