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| ===The Mixed Weibull Distribution===
| | #REDIRECT [[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:
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| <br>
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| ::<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}}}
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| </math>
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| where the value of <math>S</math> 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 <math>{{i}^{th}}</math> population, a <math>\beta_{i}</math> , or shape parameter for the <math>{{i}^{th}}</math> population and or scale parameter <math>\eta_{i}</math> for <math>{{i}^{th}}</math> 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:
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| <br>
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| ::<math>\sum_{i=1}^{s}p_{i}=1</math>
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| The mixed Weibull distribution and its characteristics are presented in more detail in Chapter [[The Mixed Weibull Distribution]].
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