Half-Normal Distribution

Author

Eric W. Weisstein
June 3, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/Half-NormalDistribution.nb.

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

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

Plot

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

Stats

Stats[HalfNormalDistribution[θ],x]//Simplify//StatsForm

Moments

Raw Momennts

FullSimplify[Integrate[2θ/Pi x^n Exp[-θ^2 x^2/Pi],{x,0,Infinity},Assumptions->θ>0&&n≥0]]

Table[(π^((1/2) (-1+n))) (θ^(-n)) Gamma[((1+n)/2)],{n,0,5}]

Table[Gamma[((1+n)/2)]Sqrt[π]^(-1+Mod[n,2]),{n,0,10}]

Through[{Numerator,Denominator}[%]]

Central Moments

Integrate[2θ/Pi (((x-(1/θ)))^n) Exp[-θ^2 x^2/Pi],{x,0,Infinity},Assumptions->θ>0&&n≥0]

Table[Integrate[2θ/Pi (((x-(1/θ)))^n) Exp[-θ^2 x^2/Pi],{x,0,Infinity},Assumptions->θ>0],{n,4}]//Factor

Through[{Mean,Variance,Skewness,KurtosisExcess}[HalfNormalDistribution[θ]]]//FullSimplify