Given two random variables X and Y, they are uncorrelated if and only if

E(XY)=E(X)E(Y)

Of course, if X and Y are independent, we also have E(XY)=E(X)E(Y). But the converse is not true.

Keep in mind: correlation can be used to describe the dependence of two variables, but not completely. However, correlation is the mostly used method. Besides, for Gaussian distributed random variables, correlation is equivalent to dependence. So just use it without any hesitate.

Here is a good example to illustrate that uncorrelated random variables can be dependent.

• Let X be a random variable uniformly distributed on [-1,1]. Random variable $Y=X^2$. Now let’s compute E(XY)-E(X)E(Y). The probability dentisty function of X is f(X)=1/2. It is easy to compute E(X)=0, E(Y)=E(X^2)=1/3, E(XY)=E(X^3)=0. So we have E(XY)-E(X)E(Y)=0. X and Y are uncorrelated. But obviously, X and Y=X^2 are not independent!!!