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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.

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

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

Multyply these inequalities to get

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

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  • $\begingroup$ 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

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