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Student[Statistics]

 CauchyRandomVariable
 Cauchy random variable

 Calling Sequence CauchyRandomVariable(a, b)

Parameters

 a - location parameter b - scale parameter

Description

 • The Cauchy random variable is a continuous probability random variable with probability density function given by:

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

 subject to the following conditions:

$a::\mathrm{real},0

 • The Cauchy random variable 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).

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):$
 > $X≔\mathrm{CauchyRandomVariable}\left(a,b\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{1}}{{\mathrm{π}}{}{b}{}\left({1}{+}\frac{{\left({u}{-}{a}\right)}^{{2}}}{{{b}}^{{2}}}\right)}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.3183098861}}{{b}{}\left({1.}{+}\frac{{\left({0.5}{-}{1.}{}{a}\right)}^{{2}}}{{{b}}^{{2}}}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{undefined}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${\mathrm{undefined}}$ (4)
 > $Y≔\mathrm{CauchyRandomVariable}\left(6,10\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 $\frac{{1}}{{2}}{+}\frac{{\mathrm{arctan}}{}\left(\frac{{1}}{{10}}{}{x}{-}\frac{{3}}{{5}}\right)}{{\mathrm{π}}}$ (5)
 > $\mathrm{CDF}\left(Y,5,\mathrm{output}=\mathrm{plot}\right)$

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.

Compatibility

 • The Student[Statistics][CauchyRandomVariable] command was introduced in Maple 18.