Chi Distribution

Author

Eric W. Weisstein
August 7, 2005

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

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

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

Special Cases

(Subscript[χ, 1])

PDF[#,x]&/@{ChiDistribution[1],HalfNormalDistribution[(Sqrt[(π/2)])]}

(Subscript[χ, 2])

PDF[#,x]&/@{ChiDistribution[2],RayleighDistribution[1]}

Plots

<<Graphics`Colors`

Show[GraphicsArray[{
Plot[Evaluate[Table[PDF[ChiDistribution[r],x],{r,10}]],{x,0,10},DisplayFunction->Identity,PlotRange->All,
PlotStyle->Hue/@Range[0,1,.09],AxesLabel->TraditionalForm/@{x,(Subscript[P, r])[x]}],
Plot[Evaluate[Table[CDF[ChiDistribution[r],x],{r,10}]],{x,0,10},DisplayFunction->Identity,
PlotStyle->Hue/@Range[0,1,.09],AxesLabel->TraditionalForm/@{x,(Subscript[D, r])[x]}]
}],ImageSize->500]

Show[GraphicsArray[ContourPlot[#[[1]][ChiDistribution[r],x],{x,.01,5},{r,.01,5},ColorFunction->Hue,FrameLabel->TraditionalForm/@{x,r},
PlotLabel->TraditionalForm[#[[2]]],
PlotPoints->100,
DisplayFunction->Identity
]&/@{{PDF,(Subscript[P, r])[x]},{CDF,(Subscript[D, r])[x]}}
],ImageSize->500]

Properties

Simplify[Stats[ChiDistribution[n],x],n∈Integers&&n>0]//StatsForm//TraditionalForm

Moments

Normalization

Integrate[PDF[ChiDistribution[n],x],{x,0,∞},Assumptions->n>1]

Raw Moments

Integrate[x^r PDF[ChiDistribution[n],x],{x,0,∞},Assumptions->n>1&&r≥0]

FullSimplify[Table[(2^(r/2))Gamma[(n+r)/2]/Gamma[n/2],{r,4}],n∈Integers&&n>1]

Central Moments

FullSimplify[Table[Integrate[(((x-(((Sqrt[2]) Gamma[((1+n)/2)])/(Gamma[(n/2)]))))^r) PDF[ChiDistribution[n],x],{x,0,∞},Assumptions->n>1],{r,4}],n>1&&n∈Integers]

v5

Timing[m=Integrate[(((x-(((Sqrt[2]) Gamma[((1+n)/2)])/(Gamma[(n/2)]))))^r) PDF[ChiDistribution[n],x],{x,0,∞},Assumptions->n>1&&r≥0&&(n|r)∈Integers]]

FullSimplify[Table[m,{r,4}],n∈Integers&&n>1]

Mean

FullSimplify[Integrate[PDF[ChiDistribution[n],x]x,{x,0,∞},Assumptions->n>1]]

Mean[ChiDistribution[n]]

Variance

FullSimplify[Integrate[PDF[ChiDistribution[n],x]x^2,{x,0,∞},Assumptions->n>1]]

FullSimplify[Variance[ChiDistribution[n]]+Mean[ChiDistribution[n]]^2]

Kurtosis Excess

KurtosisExcess[ChiDistribution[n]]//TraditionalForm

CDF

CDF[ChiDistribution[n],x]

i=Integrate[PDF[ChiDistribution[n],x],{x,0,x},Assumptions->n>1&&x>0]

FullSimplify[Subtract@@{1-((Gamma[(n/2),((x^2)/2)])/(Gamma[(n/2)])),GammaRegularized[(n/2),0,((x^2)/2)]}]