Exponential Distribution

Author

Eric W. Weisstein
May 25, 2004

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

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

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

Plot

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

Stats

Stats[ExponentialDistribution[λ],x]//StatsForm//TraditionalForm

Raw Moments

Integrate[PDF[ExponentialDistribution[λ],x]x^n,{x,0,∞},Assumptions->n≥0&&λ>0]//FullSimplify

Table[(λ^(-n)) Gamma[1+n],{n,0,4}]

FullSimplify[Gamma[n+1]-n!]

Central Moments

FullSimplify[Integrate[PDF[ExponentialDistribution[λ],x](x-(1/λ))^n,{x,0,∞},Assumptions->n≥0&&λ>0],n∈Integers&&n>0]

Table[(((λ^(-n)) Gamma[1+n,-1])/E),{n,0,6}]//FullSimplify

Skewness

Simplify[((2/(λ^3))/((((1/(λ^2))))^(3/2))),λ>0]

Skewness[ExponentialDistribution[λ]]

Kurtosis Excess

Simplify[((9/(λ^4))/((((1/(λ^2))))^2))-3,λ>0]

KurtosisExcess[ExponentialDistribution[λ]]

Characteristic Function

CharacteristicFunction[ExponentialDistribution [λ],t]

v4.2

FourierTransform[(E^(-λ x)) λ(1+Sign[x])/2,x,t]//FullSimplify

FourierTransform[(E^(-λ x)) λ UnitStep[x],x,t]//FullSimplify

InverseFourierTransform[(λ/(-I t+λ)),t,x,Assumptions->λ>0]//FullSimplify

InverseFourierTransform[(((I λ)/(t+I λ)))^n,t,x]

v5

FourierTransform[(E^(-λ x)) λ(1+Sign[x])/2,x,t]//FullSimplify

FourierTransform[(E^(-λ x)) λ UnitStep[x],x,t]//FullSimplify

FullSimplify[InverseFourierTransform[(λ/(-I t+λ)),t,x,Assumptions->λ>0]]//Timing

InverseFourierTransform[(((I λ)/(t+I λ)))^n,t,x]

v6

FourierTransform[(E^(-λ x)) λ(1+Sign[x])/2,x,t]//FullSimplify

FourierTransform[(E^(-λ x)) λ UnitStep[x],x,t]//FullSimplify

InverseFourierTransform[(λ/(-I t+λ)),t,x,Assumptions->λ>0]

InverseFourierTransform[(((I λ)/(t+I λ)))^n,t,x]

Extended Exponential Distribution

Integrate[1/β Exp[-(x-α)/β],{x,α,∞},Assumptions->β>0]

Central Moments

Integrate[x^n/β Exp[-(x-α)/β],{x,α,∞},Assumptions->α>0&&β>0&&n≥0]//FullSimplify

Table[(E^(α/β)) (β^n) Gamma[1+n,(α/β)],{n,0,4}]//FunctionExpand//FullSimplify

Raw Moments

Integrate[(x-(α+β))^n/β Exp[-(x-α)/β],{x,α,∞},Assumptions->α>0&&β>0&&n≥0]//FullSimplify

Table[(((β^n) Gamma[1+n,-1])/E),{n,0,6}]//FunctionExpand//FullSimplify

Characteristic Function

v4.2

FourierTransform[1/β Exp[-(x-α)/β]UnitStep[x-α],x,t]//FullSimplify

InverseFourierTransform[((E^(I t α))/(1-I t β)),t,x,Assumptions->β>0]//FullSimplify

v5

FourierTransform[1/β Exp[-(x-α)/β]UnitStep[x-α],x,t]

InverseFourierTransform[((E^(I t α))/(1-I t β)),t,x,Assumptions->β>0]//FullSimplify

v6

FourierTransform[1/β Exp[-(x-α)/β]UnitStep[x-α],x,t]

InverseFourierTransform[((E^(I t α))/(1-I t β)),t,x,Assumptions->β>0]

InverseFourierTransform[(((E^(I t α))/(1-I t β)))^n,t,x]