Let's say I have a RV $X$ with values $100$, $200$ and their associated probabilities, and some RV $Y$ with values $35$, $47$ and $862$ with associated probabilities.
What does it even mean to find the covariance of those Random Variables? Looking at the formula it looks like we're doing $$(x_1 - \mu_x) (y_1 - \mu_y) P(x_1 \cap y_1) + (x_2 - \mu_x) (y_2 - \mu_y) P(x_2 \cap y_2) + ... + (x_n - \mu_x) (y_n - \mu_y) P(x_n \cap y_n)$$.
How does that make sense if the random variables have different numbers of values?