Student's z-Distribution

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Code

f[z_,n_]:=Gamma[n/2]/Sqrt[Pi]/Gamma[(n-1)/2] (1+z^2)^(-n/2)
dd[z_/;z≤0,n_]:=(1/(2 (Sqrt[π])))(((Abs[z])^(1-n)) Gamma[(n/2)] Hypergeometric2F1Regularized[(1/2) (-1+n),(n/2),((1+n)/2),-(1/(z^2))])
dd[z_/;z≥0,n_]:=1-(1/(2 (Sqrt[π])))(((Abs[z])^(1-n)) Gamma[(n/2)] Hypergeometric2F1Regularized[(1/2) (-1+n),(n/2),((1+n)/2),-(1/(z^2))])

Plot

Show[GraphicsArray[{
Plot[Evaluate[Table[f[z,n],{n,2,5}]],{z,-5,5},Ticks->None,PlotStyle->List/@{Red,Yellow,Green,Blue,Violet},DisplayFunction->Identity,
AxesLabel->TraditionalForm/@{z,(Subscript[f, n])[z]}],
Plot[Evaluate[Table[dd[z,n],{n,2,5}]],{z,-5,5},Ticks->None,PlotStyle->List/@{Red,Yellow,Green,Blue,Violet},DisplayFunction->Identity,
AxesLabel->TraditionalForm/@{z,(Subscript[D, n])[z]}]
},
DisplayFunction->$DisplayFunction
],ImageSize->500,GraphicsSpacing->-.05]

Stats

FullSimplify[Integrate[f[z,n],{z,-Infinity,Infinity},Assumptions->{Re[n]>1}]]

({(Subscript[μ, 1]),(Subscript[μ, 2]),(Subscript[μ, 3]),(Subscript[μ, 4])}=FullSimplify[Table[Integrate[f[z,n] z^i,{z,-Infinity,Infinity},Assumptions->{Re[n]>i+1}],{i,4}]])//TableForm

Skewness

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

Kurtosis

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

Characteristic Function

FourierTransform[f[z,n],z,t]

Simplify[PowerExpand[%]]