represents a Weibull distribution with shape parameter α and scale parameter β.
represents a Weibull distribution with shape parameter α, scale parameter β, and location parameter μ.
- WeibullDistribution is also known as Rosin–Rammler distribution.
- The probability density for value in a Weibull distribution is proportional to for , and is zero for . »
- The probability density for value in a Weibull distribution with location parameter is proportional to for , and is zero for .
- WeibullDistribution allows α and β to be any positive real numbers and μ to be any real number.
- WeibullDistribution allows β and μ to be any quantities of the same unit dimensions, and α to be a dimensionless quantity. »
- WeibullDistribution can be used with such functions as Mean, CDF, and RandomVariate. »
Background & Context
- WeibullDistribution[α,β,μ] represents a continuous statistical distribution supported on the interval and parametrized by a real number μ (called a "location parameter") and by positive real numbers α and β (a "shape parameter" and a "scale parameter", respectively), which together determine the overall behavior of its probability density function (PDF). Depending on the values of α, β, and μ, the PDF of a Weibull distribution may have any of a number of shapes, including unimodal with a single "peak" (i.e. a global maximum) or monotone decreasing with a potential singularity nearing the lower boundary of its domain. In addition, the tails of the PDF may be "fat" (i.e. the PDF decreases non-exponentially for large values ) or "thin" (i.e. the PDF decreases exponentially for large ), depending on the values of α, β, and μ. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Weibull distribution is sometimes referred to as the Rosin–Rammler distribution, and the two-parameter form WeibullDistribution[α,β] is equivalent to WeibullDistribution[α,β,0].
- WeibullDistribution is one of four distributions (along with FrechetDistribution, ExtremeValueDistribution, and GumbelDistribution) classified under the general heading "extreme value distributions", all of which are used as tools for quantifying "extreme" or "rare" events (i.e. those that are "extremely unlikely", having datasets consisting of variates with extreme deviations from the median). The Weibull distribution is named for Swedish scientist Waloddi Weibull, though its discovery is due to Fréchet in the 1920s. Since its inception, the Weibull distribution has been used to model a number of real-world phenomena, including the distribution of particle sizes and wind speeds, as well as flood, drought, and catastrophic insurance losses. The Weibull distribution has also been used in survival analysis, manufacturing, engineering, and actuarial science.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Weibull distribution. Distributed[x,WeibullDistribution[α,β,μ]], written more concisely as xWeibullDistribution[α,β,μ], can be used to assert that a random variable x is distributed according to a Weibull distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions may be given using PDF[WeibullDistribution[α,β,μ],x] and CDF[WeibullDistribution[α,β,μ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a Weibull distribution, EstimatedDistribution to estimate a Weibull parametric distribution from given data, and FindDistributionParameters to fit data to a Weibull distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Weibull distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Weibull distribution.
- TransformedDistribution can be used to represent a transformed Weibull distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Weibull distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Weibull distributions.
- WeibullDistribution is related to several other distributions. As mentioned previously, WeibullDistribution shares qualitative relationships with ExtremeValueDistribution, FrechetDistribution, and GumbelDistribution. These relationships can be quantified by noting that the PDF of WeibullDistribution can be realized as a transformation (TransformedDistribution) of ExtremeValueDistribution, FrechetDistribution, and GumbelDistribution. WeibullDistribution is a generalization of both ExponentialDistribution and RayleighDistribution, in the sense that the CDF of WeibullDistribution[1,1/λ] and the PDF of WeibullDistribution[2, σ], respectively, are equivalent to the CDF of ExponentialDistribution[λ] and the PDF of RayleighDistribution[σ], respectively. WeibullDistribution is also closely related to MinStableDistribution, MaxStableDistribution, GammaDistribution, and GompertzMakehamDistribution.
Examplesopen allclose all
Basic Examples (5)
The lifetime of a component has WeibullDistribution with and given in . Find the probability that the component survives 300 hours:
The lifetime of a device has WeibullDistribution. Find the reliability of the device:
A component is manufactured in two factories. The products coming from factory A have a lifespan following Weibull distribution with and decades, while the time to failure for products coming from factory B follows Weibull distribution with and decades. Find the probability that a component from factory A fails before a component from factory B:
In the theory of fading channels, WeibullDistribution is used to model fading amplitude for mobile radio systems operating in the 800–900 MHz frequency range. Find the distribution of instantaneous signal-to-noise ratio where , is the energy per symbol, and is the spectral density of white noise:
Show is again a WeibullDistribution:
WeibullDistribution can be used to approximate wind speeds:
The magnitude of the annual maximum earthquake can be modeled using WeibullDistribution. Consider earthquakes in the United States in the past 200 years:
Find a stationary renewal distribution associated with a BenktanderWeibullDistribution:
Compare with a truncated WeibullDistribution:
Properties & Relations (18)
The family of WeibullDistribution is closed under minimum:
Weibull distribution is a transformation of UniformDistribution:
ExponentialDistribution is a special case of Weibull distribution:
RayleighDistribution is a special case of Weibull distribution:
Weibull distribution is a transformation of ExponentialDistribution:
FrechetDistribution is a transformation of Weibull distribution:
Weibull distribution is a special case of MinStableDistribution:
Weibull distribution is a transformation of MaxStableDistribution:
GompertzMakehamDistribution is related to Weibull distribution:
Possible Issues (3)
WeibullDistribution is not defined when either α or β is not a positive real number: