Probability Distribution Function and Shape

The t Distribution

The t-distribution was originally introduced by W.S. Gosset, whose factory prohibited him from publishing papers, so he published his writing under the name "student". Thus, the t distribution is also known as the Student- t distribution or Student's t distribution.

If Y and Z are independent random variables, Y has a ( ) distribution, and Z ~ N(0,1), then the distribution of has probability density function

It was originally developed to be used for the distribution of sample means from a normal population when the variance of the normal distribution is not known, but estimated with the sample variance. That is, if X and are the mean and variance, respectively, of a random sample of size n from a normal population with mean and variance , then has the t distribution with = n -1 degrees of freedom. Note that the t- distribution is symmetrical around t =0.

> restart:

> with(plots):

> f1:=TPDF(1,t);

> plot(f1, t=-4..4, color=black, axes=FRAMED, title="t(1) PDF");

The next bit of code produced an animation that will allow you to see what happens to the t -distribution as increases.

> for nu from 1 to 20 do

> density[nu]:=plot(TPDF(nu,x),x=-4..4):

> num:=convert(nu,string):

> tracker[nu]:=textplot([2,0.24,`nu is `.num],color=blue):

> P[nu]:=display({density[nu],tracker[nu]}):

> od:

> display([seq(P[nu], nu=1..20)], insequence=true, title="Increasing nu");