Snedecor's F-Distribution

Author

Eric W. Weisstein
November 11, 2002

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

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

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

Initialization

<<Stats`

Distribution

f[m_,n_]:=Gamma[(m+n)/2](m/n)^(m/2)F^((m-2)/2)/Gamma[m/2]/Gamma[n/2]/(1+m/n F)^((m+n)/2)

f[m,n]//TraditionalForm

Raw Moments

FullSimplify[Table[Integrate[f[m,n] F^i,{F,0,Infinity},Assumptions->{Re[m]>-2&&Re[n]>2i&&Arg[(n/m)]≠π}],{i,1,4}]]//TableForm

Central Moments

({(Subscript[μ, 1]),(Subscript[μ, 2]),(Subscript[μ, 3]),(Subscript[μ, 4])}=FullSimplify[Table[Integrate[f[m,n] (F-(n/(-2+n)))^i,{F,0,Infinity},Assumptions->{Re[m]>0&&Re[n]>2i&&Arg[(n/m)]≠π}],{i,1,4}]])//TableForm

Skewness

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

Kurtosis

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

Characteristic Functions

Simplify[PowerExpand[FourierTransform[f[m,n]UnitStep[F],F,t]]]

FullSimplify[% /.Conjugate[t]->t]