Since the formula for calculating the correlation coefficient between two RV X and Y is $E[(X-u_X)(Y-u_Y)]/σ_Xσ_Y$. I wonder that if we have a finite value for $E[(X-u_X)(Y-u_Y)]$ but found that the variance of one variable is infinite, can we conclude that the correlation is 0 because the denominator is infinite?
Like I am calculating correlation right now, $f_x$=$\frac{2}{(x+1)^3}$ and fy=$ye^{-y}$. The joint pdf is $y^2e^{-y-xy}$
I actually get the upper part $E[(X-u_X)(Y-u_Y)]$ equal to -1 whereas the variance of X is infinite. Is this situation possible? If possible, then would the correlation of $\frac{-1}{∞}$ be 0?