suppose I have two vector of unequal length, $X =\left \{ x_1, x_2, ... x_n \right \}$ and $Y =\left \{ y_1, y_2, ... y_m \right \}$ . Each element follows the same gaussian distribution , elements within the same vector are independent, however, there are $k$ disjoint pairs $\left \{ (i_1, j_1 ), (i_2,j_2), ... (i_k, j_k) \right \}$such that possible $X_{i_t} = Y_{j_t}$, by disjoint, I mean, all the $2k$ integers are distinct.
If the length are the same, I think the covariance is able to tell me some information about the dependance.
What are the statistical measure the tells me how strong they are related with respect to $k$? what test can I run to find out the number $k$ ( approximately) ? Is the problem easier if the distribution isn't Gaussian?
Thanks