Laplace Distribution

Author

Eric W. Weisstein
October 17, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Stats

Stats[LaplaceDistribution[m,b],x]//StatsForm//TraditionalForm

Plot

Show[GraphicsArray[Plot[#[LaplaceDistribution[0,1],x],{x,-5,5},AxesLabel->TraditionalForm/@{x,#[x]},
DisplayFunction->Identity,
Ticks->None,PlotRange->All,PlotStyle->Red]&/@{PDF,CDF}],
ImageSize->500
]

Properties

Characteristic Function

V5

PiecewiseExpand[((E^(((m-x) Sign[-m+x])/b))/(2 b)),(x|m)∈Reals]//FullSimplify

FourierTransform[((E^(((m-x) Sign[-m+x])/b))/(2 b)),x,t]//Timing

InverseFourierTransform[Exp[I m t]/(1+b^2t^2),t,x]

FullSimplify[%,b>0]

PiecewiseExpand[%]//FullSimplify

V7

TimeConstrained[FourierTransform[((E^(((m-x) Sign[-m+x])/b))/(2 b)),x,t],300]//Timing

InverseFourierTransform[Exp[I m t]/(1+b^2t^2),t,x]

FullSimplify[PiecewiseExpand[%],(b|m|x)∈Reals]

ExpandAll[(((E^(-((m+x)/b))) ((E^((2 x)/b)) HeavisideTheta[m-x]+(E^((2 m)/b)) HeavisideTheta[-m+x]))/(2 b))]

Moments

Sum[Binomial[n,j](((-1))^(n-j))μ'[j](μ^(n-j)),{j,0,n}]

Sum[(((-1))^(n-j))Binomial[n,j]Binomial[j,2k](b^(2k))(μ^(n-2k))Gamma[2k+1],{j,0,n},{k,0,Floor[j/2]}]

Table[Sum[(((-1))^(n-j))Binomial[n,j]Binomial[j,2k](b^(2k))(μ^(n-2k))Gamma[2k+1],{j,0,n},{k,0,Floor[j/2]}],{n,0,10}]