I have the following questions on computing the correlation coefficient.
Let us say we have two discrete random variables $X_1$ and $X_2$, where $X_1$ has $n_1$ outcomes and $X_2$ has $n_2$ outcomes. The joint is fully described by $n_1 \times n_2$ pairs. Using the standard formulae the correlation coefficient is later obtained.
I am curious about the following thing: Let us assume that out the $n_1 \times n_2$ pairs we have a subset of pairs $(x_1, x_2)$ for which the $P(X_1 = x_1, X_2 = x_2) = P(X_1 = x_1) \times P(X_2 = x_2)$. For the remaining pairs this equality does not hold.
Intuitively it looks like the remaining pairs of points are the one which will make the correlation coefficient non-zero. Is it possible to compute the correlation coefficient using only those pairs for which the equality does not hold.
Thanks, Bogdan.