If we have three iid random variables $X_1$,$X_2$,$X_3$ with common pdf f(x)= $e^{-x}$ for x greater than 0. How can we calculate the correlation coefficient between $Y_1$=$X_1$/$X_2$ and $Y_2$=$X_1+X_2$?
I know that the pdf for $Y_1$ should be $1/(1+y_1)^2$ and pdf for $Y_2$ should be $y_2e^{-y_2}$. But I do not know how to go further. Do I need to get the E($Y_1Y_2$) and variance of $Y_1,Y_2$ for this problem? Or are there any other ways to do this?
The covariance $\operatorname{Cov}(Y_1,Y_2)$ is not defined because $E(Y_1)=\int_0^\infty \frac{x}{(1+x)^2}\,dx=\infty$. As a result the correlation $\operatorname{Corr}(Y_1,Y_2)$ is also not defined.
However, one can show using a change of variables that $Y_1$ and $Y_2$ are independently distributed since their joint density factors as the product of two marginal densities. Or if you like a more statistical argument, then suppose $X_1,X_2$ are i.i.d Exponential with rate $\lambda(>0)$. The independence follows from Basu's theorem since $Y_2=X_1+X_2$ is a complete sufficient statistic for $\lambda$ and $Y_1=\frac{\lambda X_1}{\lambda X_2}$ is an ancillary statistic. Independence of $Y_1$ and $Y_2$ would have implied $\operatorname{Corr}(Y_1,Y_2)=0$ if the correlation was defined.