Let $X$ and $Y$ be continuous random variables such that the 2dimension random variable $(X,Y)$ is not continuous. why this imply that $$Cov(X,Y)>0$$
I tried to use the definition on expected value so $$Cov(X,Y)=E[XY] - E[X]E[Y]$$ $$E[XY]=\displaystyle\int\limits^{\cssId{upper-bound-mathjax}{\infty}}_{\cssId{lower-bound-mathjax}{0}} P(XY>t)dt\,\cssId{int-var-mathjax}\displaystyle-\int\limits^{\cssId{upper-bound-mathjax}{0}}_{\cssId{lower-bound-mathjax}{- \infty}} P(XY\leq t)dt\,$$
\begin{equation} E[X]E[Y]=\int^{\infty}_{-\infty} xf_x(x)dx \int^{\infty}_{-\infty}yf_y(y)dy \end{equation}
How do I procced from here? may I say that $x=y=t$?