For standard normal distribution, the expected value of $x^2$ is $1$. A natural question is that in the multivariate case, what is the expected value of $x^t\Sigma x$ for multivariate normal distribution $x \sim N(0,\Sigma)$? I have difficulty to carry out the integral, but would guess the result is related to the norm of $\Sigma$.
[Math] Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$
gaussian-integralnormal distributionprobabilitystatistics
Best Answer
Note that $x^T \Sigma x = tr(x^T \Sigma x) = tr( x x^T \Sigma)$ by invariance of trace to cyclic permutations and trace of a scalar is itself. Also, note that trace is linear (so we can bring expectations inside the trace). Since $x$ is zero mean, $E[x x^T ] = \Sigma$.
Then we have, by the aforementioned, $E[x^T \Sigma x] = E[ tr( x x^T \Sigma) ] = tr(E[x x^T \Sigma]) = tr(E[x x^T ] \Sigma) = tr(\Sigma^2)$.