BUILT-IN MATHEMATICA SYMBOL

# StableDistribution

StableDistribution[type, , , , ]
represents the stable distribution with index of stability , skewness parameter , location parameter , and scale parameter .

## DetailsDetails

• A linear combination of independent identically distributed stable random variables is also stable.
• A stable distribution is defined in terms of its characteristic function , which satisfies a functional equation where for any and there exist and such that . The general solution to the functional equation has four parameters.
• StableDistribution allows 0<≤2, , to be any real number, and to be any positive real number.
• CharacteristicFunction[StableDistribution[0, , ...], t] is continuous in and given by .
• CharacteristicFunction[StableDistribution[1, , ...], t] is discontinuous in and given by .
• is equivalent to StableDistribution[1, , 0, 0, 1].
• is equivalent to StableDistribution[1, , , 0, 1].
• StableDistribution[, , , ] is equivalent to StableDistribution[1, , , , ].
• StableDistribution can be used with such functions as Mean, CDF, and RandomVariate.

## ExamplesExamplesopen allclose all

### Basic Examples (4)Basic Examples (4)

Probability density function for type 1 for a range of skewness parameters:

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Probability density function for type 0 for various stability indexes:

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Cumulative distribution function for type 1:

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Type 0:

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Mean depends on the type:

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Variance is type independent and is only defined for :

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