I am a statistical novice trying to work something out which is probably basic. Or impossible. I have discrete probability distributions for two random variables $A$ and $B$, something like this:
| k | P(A=k) | P(B=k) |
|---|---|---|
| 0 | 0 | 0.1 |
| 1 | 0 | 0.4 |
| 2 | 0.1 | 0.4 |
| 3 | 0.15 | 0.1 |
| 4 | 0.25 | 0 |
| 5 | 0.25 | 0 |
| 6 | 0.15 | 0 |
| 7 | 0.1 | 0 |
| 8 | 0 | 0 |
| 9 | 0 | 0 |
I want to find probabilities like $P(A=k \cap B=j)$. When $A$ and $B$ are independent, I can just multiply $P(A=k)$ and $P(B=j)$. But in my setting I want to try to model situations where $A$ and $B$ are not independent.
As a very simple case, I'd like to assume that maybe there is some linear relationship between A and B, so that, say, if A ends up observed higher than expected, then B will probably also end up higher than expected. This linear relationship would have a constant built in, say $C$. I'm imagining $C$ is between -1 and 1, where 1 means they are directly correlated, 0 means they are independent, and -1 means inversely correlated.
I assume this kind of thing is well-known- what is the appropriate language and framework for me to be using? Ideally I'd like a formula for computing $P(A=k \cap B=j)$ in terms of $C$ and $P(A=k)$ and $P(B=j)$.