Prove that $\exists G(x_1), H(x_2)$ such that the joint cdf $F_{(X_1,X_2)}(x_1,x_2)=G(x_1)H(x_2)$

Question

Let $X_1,X_2$ be two real valued Random Variables. Then, $X_1,X_2$ are independent $\iff \exists G(x_1), H(x_2)$ such that the joint cdf $F_{(X_1,X_2)}(x_1,x_2)=G(x_1)H(x_2)$ .

One side is trivial : $(\implies)$ Consider, $G(x_1)=F_{X_1}(x_1)$ and $H(x_2)=F_{X_2}(x_2)$

I am having difficulty showing the other way!

I think it's enough to show that $\alpha \ne 0$ such that ,$F_{X_1} = \alpha G$ and $F_{X_2}=\frac{1}{\alpha}H$, but how?

Thanks in adavance for help!

Answer 1

Letting $x_2 \to \infty$ we get $G(x_1)=P\{X_1\leq x_1\}$. Similarly letting $x_1 \to \infty$ we get $H(x_2)=P\{X_2\leq x_1\}$. Hence $P\{X_1 \leq X_1, X_2\leq x_2\}=P\{X_1 \leq X_1\}P\{ X_2\leq x_2\}$. Can you show independence form this? I will be glad to help if you need more details.