I have two random variables $X$ and $Y$ that are jointly distributed with a distribution $F$ and affiliated to each other: $f(x,y)f(x',y')\geqslant f(x',y)f(x,y')$ for all $x,y$.
I get that this implies that higher $x$ is more likely among higher $y$'s and also vice versa.
Does this mean $F(x\mid Y=y)$ is convex in $y$ for any given $x$?
What if $Y\sim\operatorname N(0,1)$ and $X\mid Y\sim\operatorname N(Y,1)$? Then a higher $X$ is more probable among higher values of $Y$ and vice-versa (and here I'm using capital letters, for a reason, unlike what I see in the question), and \begin{align} & F(x\mid Y=y) = \Phi(x-y), \\[8pt] \text{and so } & \frac{d^2}{dy^2} F(x\mid Y=y) = -\frac d {dy} \varphi(x-y) \\[8pt] = {} & \varphi(x-y) \cdot y = \frac{-1}{\sqrt{2\pi}} \cdot e^{-(x-y)^2/2} \cdot y \end{align} and the sign of this depends on the sign of $y.$