I have the following table, where the first row are the intervals of $y$ variable and the first column are the intervals of the $x$ variable. After getting 4000 samples, I placed their count into the table according to the intervals they fell into.
\begin{array}{|c|c|c|c|} \hline x\,\&\,y& -20,-10 & -10,0 & 0,10 & 10,20\\ \hline -20,10& 0&0 &0 &500\\ \hline -10,0& 0 & 0& 1500&0\\ \hline 0,10& 0&1500 &0 &0\\ \hline 10,20& 500 & 0&0 &0\\ \hline \end{array}
We were supposed to calculate the correlation coefficient for this.
We were given the integral $$R[n_1,n_2]=\int_{-\infty}^\infty\int_{-\infty}^\infty x_1x_2p(x_1,x_2,n_1,n_2)\text{d}x_1\text{d}x_2,$$ which I believe does not make any sense (I was unable to find ANY information about it online). All I see in this equation is the expected value for the product of the two random variables. The result is -75, which again does not make sense to me, as from the table, it would seem that the expected value would be in the middle (which is $0\times 0=0$).
In my probability class, we calculated the correlation coefficient as $$r_s=\rho_{R(X),R(Y)}=\frac{\text{cov}(R(X),R(Y))}{\sigma_{R(X)}\sigma_{R(Y)}},$$ and I have a hard time trying to connect these two equations. I will appreciate any push in the right direction.
My questions are: Why isn't the expected value 0, as the probabilities are symmetric around zero? What is the connection between the double integral and the correlation coefficient formula below it?
Thank you.