0
$\begingroup$

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?

$\endgroup$
-1
$\begingroup$

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.enter image description here

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ This is great. Thank you very much. $\endgroup$ – skyline Jun 11 '15 at 13:32
  • $\begingroup$ What is the meaning of the level lines? Actually, what is this a picture of? $\endgroup$ – Did Jun 16 '15 at 20:19
  • $\begingroup$ @Did I insert an Edit to explain what the plot represent. $\endgroup$ – xecafe Jun 17 '15 at 7:53
  • $\begingroup$ Then I cannot understand why a nonzero proportion of the points are out of the triangle? $\endgroup$ – Did Jun 17 '15 at 12:02
4
$\begingroup$

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 $C$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ is the reason they are correlated simply that, $P(X=.25)$ depends on whether $Y\geq .75$? (that is, the probability that $P(X=.25)$ is either 2 or 0 depending on where we are in the triangle (and similar for other points) $\endgroup$ – user106860 May 10 '18 at 3:37
  • 1
    $\begingroup$ @user106860 Yes. One way to interpret the statement "$X$ and $Y$ are correlated" is that knowledge of the value of $Y$ affects the distribution of $X$. $\endgroup$ – grand_chat May 10 '18 at 17:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.