MATH 376 -- Probability and Statistics II

Student's   t -distributions

January 25, 2006

>    read "/home/fac/little/public_html/ProbStat0506/MSP.map";

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The following gives the form of the   t -distribution  pdf  with nu   degrees of freedom:

>    TPDF(nu,y);

GAMMA(1/2*nu+1/2)/GAMMA(1/2*nu)/(nu*Pi)^(1/2)/((1+y^2/nu)^(1/2*nu+1/2))

>    TP:=plot([y->TPDF(1,y),y->TPDF(3,y),y->TPDF(7,y)],-4..4,color=red):

>    NP:=plot(y->NormalPDF(0,1,y),-4..4,color=blue):

>    with(plots):

>    display(TP,NP);

[Maple Plot]

We see that the  t- distribution  pdf  has a shape similar to that of the standard

normal  N(0,1),  but with more probability "mass" concentrated in the tails,

and less concentrated around the mean.

As the number of degrees of freedom   nu  -> infinity ,  though, the   t- distribution

pdf approaches a standard normal:

>    plot([y->TPDF(100,y),y->NormalPDF(0,1,y)],-4..4);

[Maple Plot]

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