Probability Distribution Function and Shape

The Chi-Square Distribution

A random variable has a Chi-square distribution with degrees of freedom ( ( ) ) if and only if its probability density is given by

for x > 0

f ( x ) =

0 elsewhere.

The following code will draw the density function for the ( ) distribution for . However, you are encouraged to try other values of .

> restart;

> with(plots):

> nu:=5;

> f:=x->ChisquarePDF(nu,x);

> plot(f(x),x=0..20);

To see the effect of on the shape of the ( ) distribution, the following animation will draw a series of PDFs as varies from 1 to 21 by increments of 1. What happens as increases?

> nu[0]:=1; nu_step:=1;

> for n from 0 to 20 do

> density[n]:=plot(ChisquarePDF(nu[0]+n*nu_step,x),x=0..35):

> num:=convert(evalf(nu[0]+n*nu_step,3),string):

> tracker[n]:=textplot([18,0.3,`nu is `.num],color=blue);

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

> od:

> display([seq(P[n], n=0..20)], insequence=true);

>

This next bit of code will animate the CDF of a ( ) through the same values of as the previous example.

What happens to the CDF as increases?

> nu[0]:=1; nu_step:=1;

> for n from 0 to 20 do

> density[n]:=plot(ChisquareCDF(nu[0]+n*nu_step,x),x=0..35):

> num:=convert(evalf(nu[0]+n*nu_step,3),string):

> tracker[n]:=textplot([18,0.3,`nu is `.num],color=blue);

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

> od:

> display([seq(P[n], n=0..20)], insequence=true);

>