I was wondering if there exist any theorems that allow one to relate any joint distribution to its marginals in the form of an inequality, whether or not ##X,Y## are independent. For example, is it possible to make a general statement like this?

$$f_{XY}(x,y) \geq f_X (x) f_Y(y)$$

Also, I came across the following inequality on stackexchange (link below):

$$F_X(x) + F_Y(y) - 1 \leq F_{X,Y}(x,y) \leq \sqrt{F_X(x) F_Y(y)}$$

Is this true for all ##F##? And if so, does this inequality have a name?

Many thanks in advance!

https://stats.stackexchange.com/questions/240546/inequality-involving-joint-cumulative-and-marginal-distributions