Consider two random variables, $x$ and $y$. Denote the correlation between them by $\rho$. Assume that $E[x]$ is also a function of some parameter $\pi$ and is increasing in $\pi$. So if we increase $\pi$, we increase $E[x]$. If $\rho>0$, can I now say anything about $E[y]$? That is, if $x$ and $y$ are positively correlated and I increase $\pi$, can I deduce that $E[y]$ increases as well? Does the same hold for conditional expectation?
5
-
1$\begingroup$ Please explain what "$E[x]$ increases" might mean. Apparently it has something to do with "new information," but you haven't explained what might constitute this information, how it might be related to the random variable $(x,y),$ or how it could result in changing $x.$ $\endgroup$– whuber ♦Commented Feb 11, 2021 at 19:01
-
$\begingroup$ Made some changes, let me know if this is more clear $\endgroup$– DmlawtonCommented Feb 11, 2021 at 19:22
-
2$\begingroup$ Thank you. I would invite you to take any bivariate random variable $(x,y)$ (with finite variances) and consider the family $\{(x+\pi,y)\mid \pi\in\mathbb{R}\}.$ Then consider the related family $\{(x+\pi,y-\pi x)\mid \pi\in\mathbb{R}\}.$ What happens to $E[y]$ and $\rho(x,y)$ as $\pi$ increases? $\endgroup$– whuber ♦Commented Feb 11, 2021 at 19:36
-
$\begingroup$ I'm not exactly sure how to answer what you are asking. However im not interested in how $\rho$ changes. I'm trying to understand for a fixed positive correlation, whether increasing $\pi$ will also increase $E[y]$ $\endgroup$– DmlawtonCommented Feb 11, 2021 at 20:00
-
1$\begingroup$ The first example shows that is not the case. As another example, consider the family $\{x+\pi, y-\pi)\mid \pi\in\mathbb{R}\},$ for which any increase in $E[x]$ is matched by a concomitant decrease in $E[y].$ The point is that correlations tell you nothing whatsoever about expectations. $\endgroup$– whuber ♦Commented Feb 11, 2021 at 20:33
Add a comment
|