Template:MixedWeibullDistribution

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 [math]\displaystyle{ S }[/math] 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 [math]\displaystyle{ {{i}^{th}} }[/math] population, a [math]\displaystyle{ \beta_{i} }[/math] , or shape parameter for the [math]\displaystyle{ {{i}^{th}} }[/math] population and or scale parameter [math]\displaystyle{ \eta_{i} }[/math] for [math]\displaystyle{ {{i}^{th}} }[/math] 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 more detail in Chapter 11.