This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# WeibullDistribution

 WeibullDistributionrepresents a Weibull distribution with shape parameter and scale parameter . WeibullDistributionrepresents a Weibull distribution with shape parameter , scale parameter , and location parameter .
• 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 .
Probability density function:
With location parameter:
Cumulative distribution function:
With location parameter:
Mean:
Variance:
Median:
Probability density function:
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With location parameter:
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Cumulative distribution function:
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With location parameter:
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Mean:
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Variance:
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Median:
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 Scope   (7)
Generate a set of pseudorandom numbers that are Weibull distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a density histogram of the sample with the PDF of the estimated distribution:
Skewness depends only on the first parameter:
The limiting value:
Kurtosis depends only on the first parameter:
Kurtosis attains minimum value:
The limiting value:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
With location parameter:
Quantile function:
With location parameter:
 Applications   (7)
The lifetime of a component has WeibullDistribution with and given in . Find the probability that the component survives 300 hours:
Find the probability that the component is still working after 500 hours, after it has survived 300 hours:
Find the mean time to failure:
Simulate the failure times for 30 independent components like this:
The lifetime of a device has WeibullDistribution. Find the reliability of the device:
The hazard function is increasing in time for >1 and any value of :
Find the reliability of two such devices in a series:
Find the reliability of two such devices in parallel:
Compare the reliability of both systems for and :
A component is manufactured in two factories. The products coming from factory A have a lifespan following Weibull distribution with , , while the time to failure for products coming from factory B follows Weibull distribution with and . Find the probability that a component from factory A fails before a component from factory B:
Assume 60% of components are being manufactured in factory A. Find the distribution of the time to failure of a randomly selected component:
Find the mean time to failure:
Compare the mean time to failure for each factory origin:
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:
Find the mean:
Limiting values:
WeibullDistribution can be used to approximate wind speeds:
Find the estimated distribution:
Compare the PDF to the histogram of the wind data:
Find the probability of a day with wind speed greater than 30 km/h:
Find the mean wind speed:
Simulate daily average wind speeds for a month:
A site has mean wind speed 7 m/s and Weibull distribution with shape parameter 2:
The resulting wind speed distribution over a whole year:
The power curve for a GE 1.5 MW wind turbine:
The total mean energy produced over the course of a year is then 4.3 MWh:
The magnitude of the annual maximum earthquake can be modeled using WeibullDistribution. Consider earthquakes in the United States in the past 200 years:
Find the annual maximum:
Create a sample, eliminating the missing data:
Fit a Weibull distribution into the sample:
Compare the histogram of the sample with the PDF of the estimated distribution:
Using the model, find the probability of the annual maximum earthquake of magnitude at least 6:
Find the average magnitude of the annual maximum earthquake:
Simulate the magnitudes of the annual maximum earthquake for 30 years:
Parameter influence on the CDF for each :
Weibull distribution is closed under translation and scaling by a positive factor:
The family of WeibullDistribution is closed under minimum:
CDF of WeibullDistribution solves the minimum stability postulate equation:
Fix and and simplify:
A power of a WeibullDistribution is again a WeibullDistribution:
All moments are the same:
Compare PDFs through random sampling:
Relationships to other distributions:
The default location is 0:
Weibull distribution is a transformation of UniformDistribution:
WeibullDistribution is exponentially related to ExtremeValueDistribution:
WeibullDistribution is exponentially related to GumbelDistribution:
GumbelDistribution is a transformed Weibull distribution:
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:
WeibullDistribution is a special case of generalized GammaDistribution:
GompertzMakehamDistribution is related to Weibull distribution:
WeibullDistribution is not defined when either or is not a positive real number:
The characteristic function of the Weibull distribution does not have a closed-form representation:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: