Difference between revisions of "Lorenzian"
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| − | = Lorenzian(mode,scale) = | + | == Lorenzian(mode, scale) == |
| − | + | The [[Lorenzian]] distribution (also known as ''Cauchy, Cauchy-Lorenz, Lorenz'', and ''Breit-Wigner'') is a continuous bell-shaped distribution having the indicated «mode», and with the second parameter, «shape», specifying the half-width at the half-maximum density. | |
| − | The Lorenzian distribution (also known as Cauchy, Cauchy-Lorenz, Lorenz, and Breit-Wigner) is a continuous bell-shaped distribution having the indicated | ||
It has uses in physics, especially in the study of resonance and spectroscopy where it describes the shape of spectral lines that are broadened through various resonances. | It has uses in physics, especially in the study of resonance and spectroscopy where it describes the shape of spectral lines that are broadened through various resonances. | ||
| − | The standard form, in which | + | The standard form, in which «mode» = 0 and «shape» = 1, is known as the standard Cauchy distribution. |
| − | The Lorenz distribution has some unusual mathematical properties that are uncommon among the standard distributions. Its mean, variance and higher moments are all undefined. As a result, the law of large numbers does not apply to samples generated from a Lorenz distribution. | + | The Lorenz distribution has some unusual mathematical properties that are uncommon among the standard distributions. Its [[mean]], [[variance]] and higher moments are all undefined. As a result, the law of large numbers does not apply to samples generated from a Lorenz distribution. |
| − | One other property of interest: The ratio of two standard [[Normal|normal]] random variables follows a standard Cauchy distribution. | + | One other property of interest: The ratio of two standard [[Normal|normal]] [[random]] variables follows a standard Cauchy distribution. |
| − | = Library = | + | == Library == |
| + | Distribution Variations.ana | ||
| − | Distribution | + | ==See Also== |
| + | * [[Normal]] | ||
| + | * [[Distribution Densities Library]] | ||
Revision as of 00:56, 28 January 2016
Lorenzian(mode, scale)
The Lorenzian distribution (also known as Cauchy, Cauchy-Lorenz, Lorenz, and Breit-Wigner) is a continuous bell-shaped distribution having the indicated «mode», and with the second parameter, «shape», specifying the half-width at the half-maximum density.
It has uses in physics, especially in the study of resonance and spectroscopy where it describes the shape of spectral lines that are broadened through various resonances.
The standard form, in which «mode» = 0 and «shape» = 1, is known as the standard Cauchy distribution.
The Lorenz distribution has some unusual mathematical properties that are uncommon among the standard distributions. Its mean, variance and higher moments are all undefined. As a result, the law of large numbers does not apply to samples generated from a Lorenz distribution.
One other property of interest: The ratio of two standard normal random variables follows a standard Cauchy distribution.
Library
Distribution Variations.ana
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