Dear all,
I might just be blind, so forgive me if it is a trivial question. Given two normally distributed variables $x_1$ and $x_2$ (with zero mean), their correlation $c$ can be estimated from the samples $x_1 x_2$, $c = E[x_1 x_2]$ (where E denotes the expectation value). Now assume I want to estimate the square of the correlation, $c^2$. Unfortunately, the expectation value of $E[(x_1x_2)^2]$ for two zero-mean Gaussians is simply the product of their variances, so $E[x_1^2x_2^2] = \sigma_1^2\sigma_2^2$ where $\sigma_i^2$ is the variance of $x_i$. So how do I compute $c^2$ directly as an expectation value of independent samples?
Thanks so much!
Wieland
Best Answer
$E[x_1^2 x_2^2]=\sigma_1^2 \sigma_2^2 + 2 c^2$
(an application of Isserlis theorem)