I would like to ask for help on the proof of the following statement: $$\text{Given two random variables X and Y with joint DF}\;F_{X,Y}(x,y)$$$$\text{marginal DFs}\; F_X(x)\;\text{and}\;F_Y(y)\text{, respectively, then:}$$$$F_{X,Y}(x,y)\leq\sqrt{F_X(x)F_Y(y)}$$

I do no know how to "properly start" but I am guessing that can I use the Cauchy-Schwarz Inequality here? Although I think it is a longshot since the inequality is stated for expectations, but I t seems like one. I hope someone can help, at least on some guide on how to go about the proof. Thanks.


$0\le F_{XY}(x,y) \le F_{XY}(x,\infty)=F_X(x)$

$0 \le F_{XY}(x,y) \le F_{XY}(\infty,y)=F_Y(y)$

Multyply these inequalities to get

$(F_{XY}(x,y))^2 \le F_{X}(x)F_Y(y).$

  • $\begingroup$ Thanks a lot :) I never thought it could have been this easy. Maybe I was thinking a lot. Again, thanks. $\endgroup$ – math_stat_enthusiast Dec 31 '13 at 15:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.