# Correlation coefficient

I'm a little puzzled by the whole random variable thing.

I've got two random variables, $\mathcal{X}$ and $\mathcal{N}$, both with gaussian distribution with mean = 0 and $\sigma_{\mathcal{X}}^2$ and $\sigma_{\mathcal{N}}^2$ respectively.

The equation for correlation coefficient is

$$\rho_{XY} = E\left[\frac{X-E[X]}{\sigma_{\mathcal{X}}}\frac{Y-E[Y]}{\sigma_{\mathcal{N}}}\right]$$

I know how to find $E$ but what value do $\mathcal{X}$ and $\mathcal{N}$ actually have that I plug into the equation?

• The thing inside $[\cdot]$ is another random variable, and $E$ denotes its expectation value. You're not supposed to plug in any single values of $\mathcal{X}$ and $\mathcal{N}$.
– JiK
Commented Feb 25, 2014 at 22:16
• The trouble is that that $E$ is an expectation over $X$ and $Y$ jointly. You need, e.g., the covariance as well to be able to calculate the correlation. Commented Feb 25, 2014 at 22:40

When you calculate an expectation ($E$) of a random variable $Z$ you sum up for all values of $Z$ times the respective prbability of $Z$ taking that value. Accordingly, you work here. So every possible value of $X,Y$ will apper (be plugged) in the equation, multiplied however with the repsective probability of occuring (see formula of expected value).
If you know that two random variables are independent then that implies that they are uncorrelated. (independent implies uncorrelated but not the other way round). So in that case $$\rho_{XY}=0$$ and you do not need to apply the definition nor do any calculations to prove it.
• is the coefficient not simply zero? Since $E[\frac{XY}{\sigma_X\sigma_Y}] = 0$ Commented Feb 26, 2014 at 10:42