F-Distribution

Author

Eric W. Weisstein
December 22, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/F-Distribution.nb.

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

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

Plot

<<Graphics`Colors`

Show[GraphicsArray[{Plot[{PDF[FRatioDistribution[1,1],x],PDF[FRatioDistribution[1,2],x],PDF[FRatioDistribution[2,3],x]},{x,.01,0.99},AxesOrigin->{Automatic,0},AxesLabel->TraditionalForm/@{x,P[x]},Ticks->None,PlotStyle->{Black,Red,Blue},DisplayFunction->Identity,PlotRange->All],Plot[{CDF[FRatioDistribution[1,1],x],CDF[FRatioDistribution[1,2],x],CDF[FRatioDistribution[2,3],x]},{x,0.01,0.99},AxesLabel->TraditionalForm/@{x,"\!\(\*
StyleBox[\"D\",\nFontSlant->\"Italic\"]\)(\!\(\*
StyleBox[\"x\",\nFontSlant->\"Italic\"]\))"},Ticks->None,PlotStyle->{Black,Red,Blue},DisplayFunction->Identity,PlotRange->All]}]]

Stats

Stats[FRatioDistribution[n,m],x]//StatsForm//TraditionalForm

PDF

{PDF[FRatioDistribution[n,m],x],((Gamma[(n+m)/2](n^(n/2))(m^(m/2)))/(Gamma[n/2]Gamma[m/2]))((x^(n/2-1))/(((m+n x))^((n+m)/2)))}

FullSimplify[Subtract@@%]

Normalization

Integrate[PDF[FRatioDistribution[n,m],x],{x,0,∞},Assumptions->m>0&&n>0]

CDF

CDF[FRatioDistribution[n,m],x]

Integrate[(((m^(m/2)) (n^(n/2)) (x^(-1+(n/2))) (((m+n x))^((1/2) (-m-n))))/(Beta[(n/2),(m/2)])),{x,0,x},Assumptions->m>0&&n>0&&x>0]

FullSimplify[%70]

FullSimplify[%,m>0&&n>0&&x>0&&(m|n)∈Integers]

Characteristic Function

PowerExpand[FourierTransform[(((m^(m/2)) (n^(n/2)) (x^((1/2) (-2+n))) (((m+n x))^((1/2) (-m-n))))/(Beta[(n/2),(m/2)]))UnitStep[x],x,t]]

FullSimplify[%]

FullSimplify[%10 /.{Conjugate[t]->t,Abs[t]^2->t^2}]

Mean

Integrate[x PDF[FRatioDistribution[n,m],x],{x,0,∞},Assumptions->m>0&&n>0]

Integrate[x PDF[FRatioDistribution[n,m],x],{x,0,∞},Assumptions->m>0&&n>2]

Integrate[x PDF[FRatioDistribution[n,m],x],{x,0,∞},Assumptions->m>2&&n>0]