Computing the Correlation Coefficient of Two Random Variables

Question

I am a bit confused on computing the $\rho$ for two random variables. Let $f(X,Y) = 1$ with support given by $-x < y < x$ and $0 < x < 1$. I know that the definition of $\rho$ is simply $\rho = \frac{cov(X,Y)}{\sigma_X\sigma_Y}$. Hence, I have computed E[XY] =\begin{equation} E[XY] = \int_{0}^{1} \int_{-x}^{x} xy \,dy\,dx \end{equation}

But clearly the above quantity is just $0$, and $E[Y] = 0$. Hence, by the definition of covariance $cov(X,Y) = E[XY] - E[X]E[Y]$, but both terms in the difference are zero, hence covariance is 0 and this implies $\rho = 0$. Have I done this computation incorrectly? I feel like there is something not quite right, because there is a dependency in the support of the joint PDF, and hence it seems like there should be non-zero correlation between these random variables. Just looking for what I did wrong here if I did in fact do something wrong. Thanks for reading!

Answer 1

Your reasoning to support zero correlation is correct.

And yes, you're also correct in your assertion that $X,Y$ are dependent.

Independence implies uncorrelated, but uncorrelated doesn't imply independence.

$\;\;\;\;\;$ https://en.wikipedia.org/wiki/Independence_(probability_theory)#Expectation_and_covariance

Answer 2

Yes, correct. This is an example showing that $cov=0$ does not imply Independence.