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 $E(y^T y)$?


Thanks


Edited: Removed expectation around product.



Expectation of the sum of squares of a covariant normal samples
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 $


