BUILT-IN MATHEMATICA SYMBOL

LandauDistribution

represents a Landau distribution with location parameter

and scale parameter

.

MORE INFORMATION

The probability density for value in a Landau distribution is proportional to .25123989

LandauDistribution allows to be any real number and any positive real number.

LandauDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (3)

Probability density function:

Cumulative distribution function:

Mean and variance are not defined:

Probability density function:

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Cumulative distribution function:

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Mean and variance are not defined:

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Scope (4)

Generate a set of pseudorandom numbers that are Landau 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:

Hazard function:

Quantile function:

Applications (2)

Charged particles are bombarding a very thin target. The energy loss spectrum is then described by LandauDistribution. Define Landau function

:

Find the maximum of the Landau function:

Find the half-width of the Landau function:

For large arguments, the Landau function behaves as

:

Lindhard's first-order approximations to the Landau function:

Second-order approximation:

Properties & Relations (5)

Parameter influence on the CDF for each

:

Landau distribution is closed under translation:

Relationships to other distributions:

Landau distribution is a StableDistribution:

Evaluate the density of LandauDistribution using its defining integral representation: