Maxwell Distribution

Author

Eric W. Weisstein
May 5, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Definition

If[$VersionNumber<6,
MaxwellDistribution /: PDF[MaxwellDistribution[a_],x_]:=Sqrt[2/Pi] a^(-3) x^2 Exp[-x^2/(2a^2)];
MaxwellDistribution /:CDF[MaxwellDistribution[a_],x_]:=((2 Gamma[(3/2),0,((x^2)/(2 (a^2)))])/(Sqrt[π]));
MaxwellDistribution /:Domain[MaxwellDistribution[a_]]:=Interval[{0,Infinity}];
MaxwellDistribution /:Mean[MaxwellDistribution[a_]]:=2a Sqrt[2/Pi];
MaxwellDistribution /:Variance[MaxwellDistribution[a_]]:=(3Pi-8)a^2/ Pi;
MaxwellDistribution /:Skewness[MaxwellDistribution[a_]]:=-((2 (Sqrt[2]) (-16+5 π))/(((-8+3 π))^(3/2)));
MaxwellDistribution /:Kurtosis[MaxwellDistribution[a_]]:=3-((4 (96-40 π+3 (π^2)))/(((8-3 π))^2));
]

Stats

Stats[MaxwellDistribution[a],x]//StatsForm//TraditionalForm

Stats[MaxwellDistribution[(Sqrt[((k T)/m)])],v]//FullSimplify[#,k>0&&m>0&&T>0]&//StatsForm//TraditionalForm

Plot

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

Cumulative

CDF[MaxwellDistribution[a],x]

Integrate[PDF[MaxwellDistribution[a],t],{t,0,x}]

Simplify[FunctionExpand[2 Gamma[3/2,0,x^2/(2a^2)]/Sqrt[Pi]],a>0&&x>0]

Simplify[D[((2 Gamma[(3/2),0,((a (x^2))/2)])/(Sqrt[π])),x]>0,x>0&&a>0]

Simplify[D[-(Sqrt[a]) (E^(-((a (x^2))/2))) (Sqrt[(2/π)]) x+Erf[(((Sqrt[a]) x)/(Sqrt[2]))],x]>0,x>0&&a>0]

With[{a=1},Plot[((2 Gamma[(3/2),0,((a (x^2))/2)])/(Sqrt[π])),{x,0,5},PlotRange->All,PlotStyle->Red]]

With[{a=1},Plot[-(Sqrt[a]) (E^(-((a (x^2))/2))) (Sqrt[(2/π)]) x+Erf[(((Sqrt[a]) x)/(Sqrt[2]))],{x,0,5},PlotRange->All,PlotStyle->Red]]

Moments

Raw Moments

Simplify[RawMoment[MaxwellDistribution[a],r],a>0]

Simplify[RawMoment[MaxwellDistribution[a],#]&/@Range[5],a>0]

Central Moments

FullSimplify[CentralMoment[MaxwellDistribution[a],#]&/@Range[5],a>0]

v4.2

mu=CentralMoment[MaxwellDistribution[a],r,Assumptions->a>0]

FullSimplify[Table[mu,{r,5}]]

v5.0

CentralMoment[MaxwellDistribution[a],r]//InputForm

Mean, Variance, Skewness, Kurtosis

Simplify[{Mean[#],Variance[#],CentralMoment[#,3]/Variance[#]^(3/2),
CentralMoment[#,4]/Variance[#]^2-3}&[MaxwellDistribution[a]],a>0]

Characteristic Function

v4.2

FullSimplify[FourierTransform[PDF[MaxwellDistribution[a],x],x,t]]

InverseFourierTransform[((2 (E^(-((t^2)/(2 a))))  (a-(t^2)))/a),t,x]

v5.2

ϕ=FullSimplify[FourierTransform[UnitStep[x]PDF[MaxwellDistribution[a],x],x,t],a>0&&t∈Reals]

(Limit[D[ϕ,{t,#}],t->0,Assumptions->a>0]/I^#)&/@Range[4]

FullSimplify[Integrate[Exp[I t x]UnitStep[x]PDF[MaxwellDistribution[a],x],{x,-∞,∞},Assumptions->a>0&&t∈Reals]]

v6

FullSimplify[FourierTransform[UnitStep[x] Sqrt[2/Pi] a^(3/2) x^2 Exp[-a x^2/2],x,t,FourierParameters->{1,1}],a>0&&t∈Reals]

FullSimplify[FourierTransform[UnitStep[x] Sqrt[2/Pi] a^(3/2) x^2 Exp[-a x^2/2],x,t,FourierParameters->{1,1}],a>0&&t∈Reals]

FourierTransform[UnitStep[x] (((E^(-((x^2)/(2 (b^2))))) (Sqrt[(2/π)]) (x^2))/(b^3)),x,t,FourierParameters->{1,1}]

FullSimplify[FourierTransform[UnitStep[x] (((E^(-((x^2)/(2 (b^2))))) (Sqrt[(2/π)]) (x^2))/(b^3)),x,t,FourierParameters->{1,1},Assumptions->b>0]]

...correct...

FullSimplify[FourierTransform[UnitStep[x] (((E^(-((x^2)/(2 (b^2))))) (Sqrt[(2/π)]) (x^2))/(b^3)),x,t,FourierParameters->{1,1}],b>0&&t∈Reals]

...wrong...