The Weibull Distribution

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, [math]\displaystyle{ {\beta} \,\! }[/math]. This chapter provides a brief background on the Weibull distribution, presents and derives most of the applicable equations and presents examples calculated both manually and by using ReliaSoft's Weibull++ software.

Weibull Probability Density Function

The 3-Parameter Weibull

The 3-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 \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 2-Parameter Weibull

The 2-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 1-Parameter Weibull

The 1-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 1-parameter Weibull, we assume that the shape parameter [math]\displaystyle{ \beta \,\! }[/math] is known a priori from past experience with identical or similar products. The advantage of doing this is that data sets with few or no failures can be analyzed.