Question

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.

Answer 1

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 $