Variance of T-Student Distribution with Density Function

Date: 02/18/2005 at 06:03:46
From: Carlos 
Subject: Derivation of Variance of T-Student Distribution

I need help in deriving the variance (n/(n-2)) (n: degrees of freedom) 
of the T-Student Distribution.  The most difficult thing is how to 
apply the VAR to a ratio of a normally distributed variable divided by 
a Chi-Square.

VAR ((sqr n) z/(sqr y)), where n: degrees of freedom, z normal 
variable, y chi square.  Take out sqr n as simply n, and work with n
VAR (z/(sqr y).  From there I have no idea how to proceed.  Any help
would be much appreciated.

Date: 02/18/2005 at 13:07:01
From: Doctor George
Subject: Re: Derivation of Variance of T-Student Distribution

Hi Carlos,

Thanks for writing to Doctor Math.

Rather than work with the definition of the student T distribution it
would be better to work with its density function.  It is messy to
explain in a text format, but I'll try to explain the basic steps.

  E(T) = 0 by symmetry.

Now we need to compute E(T^2).  This is easy if you already know the
mean of the F distribution because T^2 has an F distribution.

Without making use of the F distribution we need to compute the 
integral for E(T^2).  There are two steps.

1. Apply the integration by parts technique using u = t and dv = the
remaining terms.  The u*v part of the result will be zero by symmetry.
The Integral(v*du) part will look much like the density function.

2. Carefully select a value n' as on offset from n in order to
transform the Integral(v*du) into the density function times a 
constant factor.  The new integral now equals one, and the factor 
becomes E(T^2).

Can you take it from there?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/