# Proof for X and Y independent if joint c.d.f. of both variables is the product of c.d.f of each of the two variables

The book I am reading says the following:

For any two variables $$X$$ and $$Y$$, if for every two sets $$A$$ and $$B$$ of real numbers $$Pr(X \in A \cap Y \in B) =Pr(X \in A )Pr( Y \in B)$$, then $$X$$ and $$Y$$ are independent. Further, if we can show that $$P(X \leq x \cap Y \leq y) = P(X \leq x)P(Y \leq y)$$, then we can conclude that $$X$$ and $$Y$$ are independent random variables.

Does anybody have a proof for this last statement? My book has omitted the proof.

• Have you seen a proof that the distribution function determines the distribution? That would make very short work of this problem. – whuber Mar 29 at 12:37
• @whuber I am afraid not. Could you share a link or something? – Avneesh Khanna Mar 30 at 13:03
• A modern textbook on probability or probability and measure would include a rigorous account. Just about any textbook on probability and statistics will at least assert this result and perhaps demonstrate it in simple cases. – whuber Mar 30 at 15:21