Suppose we have a joint probability density function $f(x,y)$ over the real x-y plane, and further suppose that we know that the $Cov(x,y)>0$. What I am wondering is: is there a relationship between $P(x|y=y_0)$ and $P(x|y=y_1)$ if $y_0<y_1$?
My intuitive sense is that because the covariance is positive, larger values of $x$ mean larger values of $y$, and so if you have a conditional pdf of $x$ with a lower value of $y$, that pdf is going to be more clustered around lower values of $x$, while the opposite can be said for the conditional pdf with a higher value of $y$.
And vice versa, if I have a negative covariance, then the opposite relationship holds, as lower values of $y$ are associated with higher values of $x$.
However, I am struggling to work out such a relationship more formally using the definition of covariance, which I know to be $E(XY)-E(X)E(Y)$. I also have been looking at Hoeffding's covariance identity, but no luck.
I would appreciate any hints or suggestions or references.