Properties of the Distribution

Calculation of Mean and Variance

To calculate the mean of the t distribution, we can simply integrate x *( t PDF), per the definition of mathematical expectation.

> restart;

> with(plots):

> interface(showassumed=0):

When nu=1 the integral of x*f(x) diverges so the expected value does not exist. We consider the case where nu>1 below.

> assume(nu > 1);

> f(x):=TPDF(nu,x);

> EX:=int(x*f(x),x=-infinity..infinity);

Calculating Var( X ), the variance of the t ( ) distribution. We will employ the formula: Var( X ) = E( ) -

The integral of x^2*f(x) diverges if nu <=2. We consider the case where nu>2 below.

> assume(nu > 2);

> E_X_SQ:=int((x^2)*f(x),x=-infinity..infinity);

> VarX:=simplify(E_X_SQ-EX^2);

Sp ecial Properties

As increases, the t distribution approaches the N(0,1) distribution. The next few plots and animations will try to give you a visual representation of this result.

> with(plots):

> nu:=1:

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

> n:=x->NormalPDF(0,1,x);

> plot([f(x),n(x)],x=-5..5,title="t(1) with N(0,1)");

The red plot is the t (1) distribution, and the green plot is the N(0,1) distribution.

The following animation portrays how the t approaches the standard normal distribution.

> for nu from 1 to 20 do

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

> num:=convert(nu,string):

> standnorm:=plot(n(x),x=-4..4,color=black):

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

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

> od:

> display([seq(P[nu], nu=1..20)], insequence=true, title="t gets close to N(0,1) as nu increases");

>