Rayleigh Distribution

Author

Eric W. Weisstein
October 11, 2005

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

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

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

Plot

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

Stats

Stats[RayleighDistribution[r],x]//StatsForm//TraditionalForm

PDF

Integrate[PDF[RayleighDistribution[r],x],{x,0,∞},Assumptions->r>0]

CDF

Integrate[PDF[RayleighDistribution[r],x],{x,0,s},Assumptions->r>0]

Raw Moments

v5.1

Integrate[x^n (((E^(-(x^2)/(2(s^2)))) x)/(s^2)),{x,0,∞},Assumptions->s>0&&n≥0]

(Overscript[μ, ′])[n_]:=(2^(n/2)) (s^n) Gamma[1+(n/2)]

Table[Integrate[x^n (((E^(-(x^2)/(2(s^2)))) x)/(s^2)),{x,0,∞},Assumptions->s>0],{n,0,4}]

Central Moments

v5.1

With[{μ=(Sqrt[(π/2)]) s},
Integrate[(((x-μ))^n) (((E^(-(x^2)/(2(s^2)))) x)/(s^2)),{x,0,∞},Assumptions->s>0&&n≥0]
]

Table[With[{μ=(Sqrt[(π/2)]) s},
Integrate[(((x-μ))^n) (((E^(-(x^2)/(2(s^2)))) x)/(s^2)),{x,0,∞},Assumptions->s>0]],{n,4}]

v5.2

FullSimplify[With[{μ=(Sqrt[(π/2)]) s},
Integrate[(((x-μ))^n) (((E^(-(x^2)/(2(s^2)))) x)/(s^2)),{x,0,∞},Assumptions->s>0&&n≥0&&n∈Integers]
],s>0&&n≥0&&n∈Integers]

FullSimplify[FunctionExpand[%83],n∈Integers&&n>0]

Table[(2^(-(3/2) (2+n))) (((-s))^n) ((((-1))^n) (4^(1+n)) n (Gamma[(n/2)] Hypergeometric1F1[(1/2)-(n/2),(1/2),-(π/4)]-(Sqrt[π]) Gamma[((1+n)/2)] Hypergeometric1F1[1-(n/2),(3/2),-(π/4)])+(π^((3+n)/2)) n! HypergeometricPFQRegularized[{1,(3/2)},{((3+n)/2),((4+n)/2)},-(π/4)]),{n,4}]//FullSimplify

Mean, Etc.

Skewness

(Subscript[μ, 3])/((Subscript[μ, 2])^(3/2))

Kurtosis

(Subscript[μ, 4])/((Subscript[μ, 2])^2)-3

Cumulants

{(Overscript[μ, ′])[1],(Overscript[μ, ′])[2]-(((Overscript[μ, ′])[1])^2),2(((Overscript[μ, ′])[1])^3)-3(Overscript[μ, ′])[1](Overscript[μ, ′])[2]+(Overscript[μ, ′])[3]}//Simplify//ColumnForm

Characteristic Function

v5.0.1

FourierTransform[(((E^(-((x^2)/(2 (s^2))))) x)/(s^2))UnitStep[x],x,t]

v5.1

FourierTransform[(((E^(-((x^2)/(2 (s^2))))) x)/(s^2))UnitStep[x],x,t]

FourierTransform[(((E^(-((x^2)/(2(s^2))))) x)/(s^2))UnitStep[x],x,t,Assumptions->s>0]

FullSimplify[%]

CoefficientList[Series[Log[1-(E^(-(1/2) (s^2) (t^2))) (Sqrt[(π/2)]) s t (-I+Erfi[((s t)/(Sqrt[2]))])],{t,0,5}],t]Table[n!/I^n,{n,0,5}]//Simplify//ColumnForm

Check  with +I

CoefficientList[Series[Log[1-(E^(-(1/2) (s^2) (t^2))) (Sqrt[(π/2)]) s t (I+Erfi[((s t)/(Sqrt[2]))])],{t,0,5}],t]Table[n!/I^n,{n,0,5}]//Simplify//ColumnForm