the factorization criterion for independence

Checking independence from the definition — testing every pair of ranges — is hopeless in practice. The factorization criterion gives a workable shortcut: just look at whether the joint pmf or pdf splits cleanly into a part that depends only on x times a part that depends only on y.

The clean version: X and Y are independent if and only if the joint factors as the product of the marginals everywhere — p(x, y) = p_X(x) p_Y(y) in the discrete case, or f(x, y) = f_X(x) f_Y(y) in the continuous case. There is also a handy looser form: X and Y are independent precisely when the joint density can be written as some function of x alone times some function of y alone, f(x, y) = g(x) h(y), with the SAME factorisation holding over a region that is itself a rectangle (the support must be a product set). The two leftover constants then sort themselves into the proper marginals.

Two warnings travel with this rule. First, the support has to factor too: if x and y are tied together by the region they live in — say 0 < x < y < 1, where the allowed range of one depends on the other — they are dependent no matter how the formula looks. Second, factorisation is the test for independence itself, which is far stronger than zero covariance; passing this test rules out every kind of dependence, not just the linear kind.