Analyzing math theory of random variables. Something fishy messes with my logic. Two questions:
When we have two random variables $X$ and $Y$, to get covariance $Cov (X, Y)$ is basically to get how the values of X and Y move relative to each other. Is this true?
Because each random variable may generate potentially infinite sets of values, how can we determine the covariance over the infinite sets?
Just to recall the definition I found:
$Cov(X,Y)=\mathbb E\big[(X-\mathbb E[X])(Y-\mathbb E[Y])\big]=\mathbb E[XY]-(\mathbb E[X])(\mathbb E[Y])$
Uses expected values to define the covariance. This looks OK.