Noncentral Student's t-Distribution

Author

Eric W. Weisstein
September 23, 2005

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

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

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

Initialiation

Statistics

Stats[NoncentralStudentTDistribution[n,λ],x]//StatsForm//TraditionalForm

PDF

PDF[NoncentralStudentTDistribution[n,λ],x]//TraditionalForm

CDF

v4.0

CDF[NoncentralStudentTDistribution[n,λ],x]//Timing

v4.2

CDF[NoncentralStudentTDistribution[n,λ],x]//Timing

v5.0.1

CDF[NoncentralStudentTDistribution[n,λ],x]//Timing

v5.1.0

CDF[NoncentralStudentTDistribution[n,λ],x]//Timing

v5.2

CDF[NoncentralStudentTDistribution[n,λ],x]//Timing

v6

CDF[NoncentralStudentTDistribution[n,λ],x]//Timing

Moments

v5.2

Table[Integrate[x^k PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}],{k,0,2}]//Timing

Normalization

v5.2

Integrate[PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}]//Timing

v6

Integrate[PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}]//Timing

Mean

Mean[NoncentralStudentTDistribution[n,λ]]

Integral

v5.2

Integrate[x PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}]//Timing

v6

Integrate[(x^( ))PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}]//Timing

Variance

Variance[NoncentralStudentTDistribution[n,λ]]

var=FullSimplify[Variance[NoncentralStudentTDistribution[n,λ]],n∈Integers&&n>1]

Table[var,{n,3,6}]//FullSimplify

Integral

v5.2

Integrate[(x^(2 ))PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}]//Timing

v6

Integrate[(x^(2 ))PDF[NoncentralStudentTDistribution[n,λ],x],{x,0,∞}]//Timing

Skewness

Skewness[NoncentralStudentTDistribution[n,λ]]

Simplify[Skewness[NoncentralStudentTDistribution[n,λ]],n>1&&n∈Integers]

TraditionalForm[%]

Kurtosis Excess

KurtosisExcess[NoncentralStudentTDistribution[n,λ]]

MyCollect[x_,y_]:=Module[{c=CoefficientList[x,y]},y^Range[0,Length[c]-1].Factor[c]]

Simplify[Expand[KurtosisExcess[NoncentralStudentTDistribution[n,λ]]],n>1&&n∈Integers]

TraditionalForm[%19]