MATH 376 -- Probability and Statistics II 

Student's  t-distributions 

January 26, 2010 

 

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

 

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1728983 (1)
 

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),... (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);
 

Plot_2d
 

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);
 

Plot_2d
 

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