# Is conditional expectation evaluated by the copula strictly increasing when the correlation coefficient is positive and vice versa?

I used the copula to evaluate the $$\mathbb{E}[Y|X]$$ and from my experiments on some copulas, I observed that when the random variables have positive correlation coefficient, $$\mathbb{E}[Y|X]$$ is strictly increasing and when they are negatively correlated $$\mathbb{E}[Y|X]$$ is strictly decreasing. Also, the conditional expectation plots in this post show the strict behavior too. I'm wondering if this is true and how to prove it.

For example, let $$c=\Phi(1)\approx 0.8413$$ where $$\Phi$$ is the standard normal cdf. Consider the bivariate copula that corresponds to $$V=1-U$$ for $$1-c and $$V=U$$ otherwise, where $$U$$ is standard uniform. The correlation of $$U$$ and $$V$$ is positive (roughly 0.36), but $$E(V|U=u)$$ is decreasing when $$u$$ is in that middle section.
Now let $$X=\Phi^{-1}(U)$$ and $$Y=\Phi^{-1}(V)$$; $$(X,Y)$$ has the same copula. Now for $$(X,Y)$$ the Pearson correlation is higher (around 0.6) but again the conditional expectation function is not strictly increasing.
The value of $$c$$ is not special; I simply chose it because it was fairly clear that the correlation in the Gaussian margins case would be positive. For the copula itself (uniform margins), any positive $$c$$ less than about 0.8968 works, for the Gaussian margins case positive $$c$$ less than about 0.938 works. You could choose a convenient value of $$c$$ and compute an exact correlation on the copula if you wanted.