Student[Statistics]
CauchyRandomVariable
Cauchy random variable
Calling Sequence
Parameters
Description
Examples
References
Compatibility
CauchyRandomVariable(a, b)
a
-
location parameter
b
scale parameter
The Cauchy random variable is a continuous probability random variable with probability density function given by:
ft=1πb1+t−a2b2
subject to the following conditions:
a::real,0<b
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).
withStudentStatistics:
X≔CauchyRandomVariablea,b:
PDFX,u
1πb1+u−a2b2
PDFX,0.5
0.3183098861b1.+0.5−1.a2b2
MeanX
undefined
VarianceX
Y≔CauchyRandomVariable6,10:
PDFY,x,output=plot
CDFY,x
12+arctanx10−35π
CDFY,5,output=plot
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.
The Student[Statistics][CauchyRandomVariable] command was introduced in Maple 18.
For more information on Maple 18 changes, see Updates in Maple 18.
See Also
Statistics[Distributions][Cauchy]
Student
Student[Statistics][RandomVariable]
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