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



Expression for the square of the correlation of two Gaussian variables as an expectation value
$E[x\_1^2 x\_2^2]=\sigma\_1^2 \sigma\_2^2 + 2 c^2$


(an application of [Isserlis theorem](http://en.wikipedia.org/wiki/Isserlis_theorem))


