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Statistics[Distributions][Cauchy] - Cauchy distribution

Calling Sequence

Cauchy(a, b)

CauchyDistribution(a, b)

Parameters

a

-

location parameter

b

-

scale parameter

Description

• 

The Cauchy distribution is a continuous probability distribution with probability density function given by:

ft=1πb1+ta2b2

  

subject to the following conditions:

a::real&comma;0<b

• 

The Cauchy distribution does not have any defined moments or cumulants.

• 

The Cauchy variate Cauchy(a,b) is related to the standardized variate Cauchy(0,1) by Cauchy(a,b) ~ a + b * Cauchy(0,1).

• 

The ratio of two independent unit Normal variates N and M is distributed according to the standard Cauchy variate: Cauchy(0,1) ~ N / M

• 

The standard Cauchy variate Cauchy(0,1) is a special case of the StudentT variate with one degree of freedom: Cauchy(0,1) ~ StudentT(1).

• 

Note that the Cauchy command is inert and should be used in combination with the RandomVariable command.

Examples

withStatistics&colon;

X:=RandomVariableCauchya&comma;b&colon;

PDFX&comma;u

1&pi;b1&plus;ua2b2

(1)

PDFX&comma;0.5

0.3183098861b1.&plus;0.51.a2b2

(2)

MeanX

undefined

(3)

VarianceX

undefined

(4)

See Also

Statistics, Statistics[Distributions], Statistics[RandomVariable]

References

  

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

  

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

  

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics.6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.


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