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Given a series of samples $y_1, y_2, \ldots, y_n \sim N(0,\Sigma)$, I'm looking to find the expection, $E(y^T y)$. It's fairly easy to show that $y^T \Sigma^{-1} y \sim \chi^2$, but then I get stuck. Maybe there isn't a closed form solution for $E(y^T y)$?


Edited: Removed expectation around product.

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up vote 3 down vote accepted

There isn't a closed form solution for the distribution, but there is for the expectation. It's called a Quadratic Form: http://en.wikipedia.org/wiki/Quadratic_form_(statistics)

$E(y^T y) = tr(\Sigma)$

Also, I think you mean to say that $y^T \Sigma^{-1} y \sim \chi ^2$, not $E(y^T \Sigma^{-1} y) \sim \chi ^2 $

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Thanks for the pointer! Never heard of quadratic forms before - some interesting results. –  James Philbin Apr 27 '11 at 17:43

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