Weibull Distribution

The Weibull distribution, also known as the Rosin-Rammler distribution, is a continuous probability distribution used to model the lifetime of equipment in reliability engineering. It is characterized by its scale and shape parameters.

The Weibull distribution has a scale parameter $\eta$ and a shape parameter $\beta$. The probability density function (PDF) is given by:

$$f(x;\beta ,\eta )=\frac{\beta }{\eta }{\left(\frac{x}{\eta }\right)}^{\beta -1}{e}^{-(x/\eta {)}^{\beta }}$$

The cumulative distribution function (CDF) is:

$$F(x;\beta ,\eta )=1-e^{-(x/\eta {)}^{\beta }}$$

The Weibull distribution is defined for all non-negative values (x ≥ 0), with both the shape parameter β and scale parameter η required to be strictly positive.

• Reliability engineering uses it to model component lifetimes and failure rates.

• Wind energy assessment employs it to model wind speed distributions.

• Materials science uses it for analyzing material strength and fatigue life.

• Weather forecasting applies it to extreme weather event prediction.

Notable properties include:

• The hazard rate function is $h(x)=\frac{\beta }{\eta }(\frac{x}{\eta }{)}^{\beta -1}$.

• The characteristic function has no closed form expression.

• The distribution exhibits different failure rate behaviors depending on $\beta$: decreasing ($\beta <1$), constant ($\beta =1$), or increasing ($\beta >1$).

• When $\beta =1$, the Weibull distribution reduces to the exponential distribution.

• When $\beta =2$, it becomes the Rayleigh distribution.

• When $\beta =3.4$, it approximates the normal distribution.

• The Weibull distribution is a special case of the generalized extreme value distribution.

The Weibull distribution is implemented by the WeibullDistribution class. It has one constructor with two arguments. The first argument is the location parameter, and corresponds to the mode of the probability density function. The second argument is the scale parameter.

The following constructs the same Weibull distribution with scale parameter 6.8 and shape parameter 4.1:

var weibull = new WeibullDistribution(6.8, 4.1);

Dim weibull = New WeibullDistribution(6.8, 4.1)

No code example is currently available or this language may not be supported.

let weibull = WeibullDistribution(6.8, 4.1)

The WeibullDistribution class has three specific properties, LocationParameter, ScaleParameter and ShapeParameter, which return the parameters of the distribution.

WeibullDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the location and scale parameters of the distribution.

var random = new Pcg32();
double sample = WeibullDistribution.Sample(random, 6.8, 4.1);

Dim random = New Pcg32()
Dim sample = WeibullDistribution.Sample(random, 6.8, 4.1)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = WeibullDistribution.Sample(random, 6.8, 4.1)

The above example uses the Pcg32 class to generate uniform random numbers.

For more information on the Weibull distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 21. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 46. Wiley Series in Probability and Statistics.

• Weibull, W. (1951). "A statistical distribution function of wide applicability". Journal of Applied Mechanics.

• Nelson, W. (1982). "Applied Life Data Analysis". John Wiley & Sons.

Other Resources