From: fisico32 on
hello forum.

two (or more) random variables x and y are totally independent if their
joint pdf can be factorized: f(x,y)=f(x)*f(y).
Also E[g1(X)g2(Y)]=E[g1(X)]E[g2(Y)], correct? Why does the fact that any
functions g1() and g2() make the variables independent, while
E[X Y]=E[X]E[Y] makes them only uncorrelated?

If the pdf is not factorizable but symmetric, then they are uncorrelated
(linearly independent only)but dependent.

What does it mean if the joint pdf is factorizable but not symmetric? IS
that possible? Is there any type of dependence going on or just complete
independence because of the factorization?

Also, what is the difference between two dependent random variables and two
mutually conditionally dependent random variables?