Uncorrelatedness and independence

Two random variables $X$ and $Y$ are independent if their joint PDF $f_{XY}(x,y)$ can be separated into a product of their individual PDFs:

$$ f_{XY}(x,y) = f_X(x) f_Y(y) $$

For any functions $g(\dot)$ and $h(\dot)$:

$$ E[g(X)h(Y)] = E[g(X)]E[h(Y)] $$

Two random variables $X$ and $Y$ are independent if:

$$ E[XY] = E[X]E[Y] $$

Alternatively:

$$ Cov(X, Y) = 0 $$

Independence implies uncorrelatedness, but uncorrelatedness does not imply independence.