Suppose $X$ and $Y$ are positive valued continuous random variables. Is it possible to find a non-linear function $f$, such that

independence between $Y$ and $X+\frac{X}{Y}$ implies independence between $Y$ and $X+f\left(\frac{X}{Y}\right)$?

I am guessing it is not possible. Trying to find a counter example, to show if the independence to be hold then $f$ must be a linear function. For showing this I am using product moment correlation, which leads covariance of the two part are same. But covariance same may not imply that the transformation is linear. Here I stuck.


closed as unclear what you're asking by whuber Dec 13 '15 at 15:31

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  • 1
    $\begingroup$ What is $W$? Another random variable? A typo for $X$? It would be interesting to consider whether it is possible for $Y$ and $X+\frac XY$ to be independent random variables at all. Do you have some joint distribution for $X$ and $Y$ in mind where this condition holds? $\endgroup$ – Dilip Sarwate Dec 13 '15 at 14:56
  • $\begingroup$ Until the issues raised by @Dilip are cleared up, there is not an answerable question here. $\endgroup$ – whuber Dec 13 '15 at 15:32
  • $\begingroup$ @DilipSarwate. Thanks for pointing out the typo. $W$ is typo for $Y$. I don't have any joint distribution in my mind for this problem. Actually this question arises from a real scientific problem. $\endgroup$ – Janak Dec 30 '15 at 11:03
  • $\begingroup$ Unless you have an example of positive continuous random variables $X$ and $Y$ such that $Y$ and $X+\frac XY$ are independent random variables, there is no point in trying to prove that $$Y~\text{and}~X+\frac XY~\text{independent}~\Longrightarrow Y~\text{and}~X+f\left(\frac XY\right)~\text{independent}.$$ Starting from a false hypothesis, one can prove any statement, true or false. $\endgroup$ – Dilip Sarwate Dec 31 '15 at 4:43