Six Sigma and Beyond: Design for Six Sigma, Volume VI
One of the more popular models for time-to-failure (TTF), Weibull distributions take many shapes and are typically identified as in the following illustration.
Weibull probability density function (pdf)
Cumulative distribution
Two parameters:
| Shape parameter: | a | (changes shape not scale) |
| Scale parameter: |
| (changes scale not shape) |
Some authors define = 1/ · and a = ²
In a typical Weibull distribution shown below, there are some general characteristics
-
Mean:
-
Variance:
-
1/ also referred to as "characteristic life" or "time constant," the life or time at which 63.2% of population has failed.
-
If a = 1, the Weibull reduces to the exponential distribution.
-
If a = 2, the Weibull reduces to the Rayliegh distribution.
-
If a ‰ˆ 3.5, the Weibull approximates the normal distribution.
-
For a < 1, reliability function decays less rapidly .
-
For a > 1, reliability function decays more rapidly.
-
A useful model for the failure time (or length of life) distributions of produces and processes.
-
Does not assume that the failure rate, , is a constant as do the Exponential and Gamma distributions.
-
Has the advantage that the distribution parameters can be adjusted to fit many situations; because of this adaptability it is widely used in reliability engineering.
-
The cumulative distribution has closed form expression that can be used to compute areas under the Weibull curve.
-
Estimates of the two parameters, and a, can be obtained when ranked sample data are plotted on scale adjusted cumulative percentile (See Probability Plots).
| Note | Characteristic life t = 1/ corresponds to the 63.2% |
-
Weibull reliability or survival function:
-
Weibull failure distribution: (same as cumulative distribution)
-
Weibull hazard rate function:
-
The shape parameter a, can be used to adjust the shape of the Weibull distribution to allow it to model a great many life (time) related distributions found in engineering.
THREE-PARAMETER WEIBULL DISTRIBUTION
If failures do not have the possibility of starting at t = 0, but only after a finite time t O , a time-shift variable can be used to redefine the Weibull reliability function:
R ( t ) =
where the time t O is called the failure free time or minimum life.