  Measuring strength or degree of statistical dependence between two random
variables is a common problem in many domains. Pearson's correlation
coefficient $\rho$ is an accurate measure of linear dependence. We show that
$\rho$ is a normalized, Euclidean type distance between joint probability
distribution of the two random variables and that when their independence is
assumed while keeping their marginal distributions. And the normalizing
constant is the geometric mean of two maximal distances, each between the joint
probability distribution when the full linear dependence is assumed while
preserving respective marginal distribution and that when the independence is
assumed. Usage of it is restricted to linear dependence because it is based on
Euclidean type distances that are generally not metrics and considered full
dependence is linear. Therefore, we argue that if a suitable distance metric is
used while considering all possible maximal dependences then it can measure any
non-linear dependence. But then, one must define all the full dependences.
Hellinger distance that is a metric can be used as the distance measure between
probability distributions and obtain a generalization of $\rho$ for the
discrete case.
