Cauchy Distribution

Author

Eric W. Weisstein
September 13, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/CauchyDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CauchyDistribution.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plot

Show[
GraphicsArray[Block[{$DisplayFunction=Identity},Plot[#[CauchyDistribution[0,1],x],{x,-10,10},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,#[x]},Ticks->None,
PlotRange->All]&/@{PDF,CDF}]
],ImageSize->500
]

Stats

Stats[CauchyDistribution[μ,b],x]//StatsForm

Normalization

Integrate[b/(b^2+(x-μ)^2)/Pi,{x,-∞,∞},Assumptions->{b>0,μ>0}]//Timing

CDF

Integrate[b/(b^2+(x-μ)^2)/Pi,{x,0,∞},Assumptions->{b>0,μ>0}]//Timing

Moments

Plot[x/(1+x^2),{x,-70,70}]

Table[Integrate[x^n b/(b^2+(x-μ)^2)/Pi,{x,-Infinity,Infinity},Assumptions->{b>0,μ>0}],{n,0,4}]//ColumnForm

Characteristic Function

ϕ=FourierTransform[(b/(π ((b^2)+(((x-μ))^2)))),x,t,Assumptions->{μ>0&&b>0}]

FullSimplify[ϕ,t>0]

FullSimplify[ϕ,t<0]

FullSimplify[ϕ,t==0]

FullSimplify[diff=Subtract@@{(E^(-t (b-I μ))) ((E^(2 b t)) UnitStep[-t]+UnitStep[t]),Exp[I μ t-b Abs[t]]}]

PiecewiseExpand[diff,t==0]

diff/.t->0

Limit[diff,t->0,Direction->#]&/@{-1,1}

InverseFourierTransform[Exp[I μ t-Γ Abs[t]/2],t,x]//FullSimplify//Timing

InverseFourierTransform[(Exp[I μ t-Γ Abs[t]/2])^n,t,x]//FullSimplify//Timing