Proof of independence of random variables X and Y

Question

Prove that random variables X and Y are independent if and only if FX,Y(x,y) = FX(x)FY(y)

Answer 1

Consider the definition of independence:

1) F_X(X | Y) = F_X(X)

That is, having information about Y does not affect the possible outcomes for X.

Next, apply the definition of a conditional probability:

2) F_X(X | Y) = F_X,Y(X, Y) / F_Y(Y)

If you combine equations 1 and 2, you can obtain F_X,Y(X, Y) = F_X(X) * F_Y(Y).