Normally Distributed and Uncorrelated Does Not Imply Independent

In probability theory, two random variables being uncorrelated does not imply their independence. In some contexts, uncorrelatedness implies at least pairwise independence (as when the random variables involved have Bernoulli distributions).

It is sometimes mistakenly thought that one context in which uncorrelatedness implies independence is when the random variables involved are normally distributed. However, this is incorrect if the variables are merely marginally normally distributed but not jointly normally distributed.

Suppose two random variables X and Y are jointly normally distributed. That is the same as saying that the random vector (X, Y) has a multivariate normal distribution. It means that the joint probability distribution of X and Y is such that for any two constant (i.e., non-random) scalars a and b, the random variable aX + bY is normally distributed. In that case if X and Y are uncorrelated, i.e., their covariance cov(X, Y) is zero, then they are independent. However, it is possible for two random variables X and Y to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples are given below.