Independence, conditioning, and correlations

Question

Suppose $X$ and $Y$ are independent random variables uniformly distributed on $[0,1]$. Suppose we consider a conditional distribution of $X$ and $Y$ on some event $C$. Is it possible that these conditional distributions are correlated?

Answer 1

An image generated with Python function seaborn.jointplot http://stanford.edu/~mwaskom/software/seaborn/generated/seaborn.jointplot.html, illustrating the answer given by @grand.chat.

EDITED The graph of a joint probability density function of $(X,Y)$ is a surface $z=f_{X,Y}(x,y)$. Usually to illustrate such a distribution one displays its contour plot, that is some level lines projected onto the $x-y$ plane.

In our case the joint p.d.f. $f_{X,Y|((X,Y)\in C)}$ is equal to $2$ for $(x,y)\in C$ and zero otherwise. Hence the theoretical contour plot consists in just one contour line (the triangle of vertices $(0,0), (0,1), (1,1)$). But in the figure posted here it is drawn the contour plot of the estimated from data joint distribution function (I simulated $(X,Y)|((X,Y)\in C)$). Since this experimental joint pdf is not a constant at the points of $C$, you can see more contours lines, not just one.

Answer 2

Yes. Try the event $C:=\{Y>X\}$. The joint density of $(X,Y)$ conditional on $C$ is uniform on the 'upper triangle' of the unit square, so $Y$ and $X$ are positively correlated given $C$.