WeibullDistribution

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:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

With location parameter:

Quantile function:

With location parameter:

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:

Find the amount of fading:

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 truncated WeibullDistribution:

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: