The Cauchy distribution is a continuous probability distribution with probability density function given by:
subject to the following conditions:
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.
X ≔ RandomVariable⁡Cauchy⁡a,b:
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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