Edited by author 10-17-2002 12:35 PM
Uncorrelated means that the covariance E[[X-Ex][Y-Ey]] for X,Y is zero. Independence means that their densities can be factored into pieces which only depend on X and Y respectivly.

Two variables X and Y are uncorrelated if
E[[X-EX][Y-EY]] = E[XY - XEY - YEX - EXEY]
                = E[XY] - EXEY
                = 0

i.e. EXY = EXEY

The requirement for independence is much stronger

it is

Eg_1(X)g_2(Y) = Eg_1(X)Eg_2(Y)

where g_1 and g_2 are arbitrary measurable (read functions which have a finite integrals) functions.

hence, independence implies zero correlation but not the otherway around.