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.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Cauchy}\left(a\,b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\frac{1}{\mathrm{\pi }b\left(1+\frac{{\left(u-a\right)}^2}{b^2}\right)}$

$\mathrm{PDF}\left(X\,0.5\right)$

$\frac{0.3183098861}{b\left(1.+\frac{{\left(0.5-1.a\right)}^2}{b^2}\right)}$

$\mathrm{Mean}\left(X\right)$

$\mathrm{undefined}$

$\mathrm{Variance}\left(X\right)$

$\mathrm{undefined}$

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.