Variance of T-Student Distribution with Density FunctionDate: 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/ |
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