Template:Weibull Probability Density Function

Weibull Probability Density Function

The Three-Parameter Weibull Distribution

The three-parameter Weibull pdf is given by:

[math]\displaystyle{ f(T)={ \frac{\beta }{\eta }}\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta -1}e^{-\left( {\frac{T-\gamma }{\eta }}\right) ^{\beta }} }[/math]

where,

[math]\displaystyle{ f(T)\geq 0,\text{ }T\geq 0\text{ or }\gamma, }[/math]

[math]\displaystyle{ \beta\gt 0\ \,\! }[/math],

[math]\displaystyle{ \eta \gt 0 \,\! }[/math],

[math]\displaystyle{ -\infty \lt \gamma \lt +\infty \,\! }[/math]

and,

[math]\displaystyle{ \eta= \,\! }[/math] scale parameter, or characteristic life

[math]\displaystyle{ \beta= \,\! }[/math] shape parameter (or slope),

[math]\displaystyle{ \gamma= \,\! }[/math] location parameter (or failure free life).

The Two-Parameter Weibull Distribution

The two-parameter Weibull pdf is obtained by setting [math]\displaystyle{ \gamma=0 \,\! }[/math], and is given by:

[math]\displaystyle{ f(T)={ \frac{\beta }{\eta }}\left( {\frac{T}{\eta }}\right) ^{\beta -1}e^{-\left( { \frac{T}{\eta }}\right) ^{\beta }} \,\! }[/math]

The One-Parameter Weibull Distribution

The one-parameter Weibull pdf is obtained by again setting [math]\displaystyle{ \gamma=0 \,\! }[/math] and assuming [math]\displaystyle{ \beta=C=Constant \,\! }[/math] assumed value or:

[math]\displaystyle{ f(T)={ \frac{C}{\eta }}\left( {\frac{T}{\eta }}\right) ^{C-1}e^{-\left( {\frac{T}{ \eta }}\right) ^{C}} \,\! }[/math]

where the only unknown parameter is the scale parameter, [math]\displaystyle{ \eta\,\! }[/math].

Note that in the formulation of the one-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience on identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.