Modelling dependence between two random variables
There are a few ways to model/represent the dependence between two random variables. For example there is the linear correlation or one can write one random variable as a function of the other. Then there is also copula functions that can be used to model dependence between two variables.
I was wondering if there is another way to model/represent the dependence between two random variables other than the three I've mentioned above? Is there another way to make two random variables mathematically dependent?
I've been thinking about this for quite a while now so help is much appreciated.
Re: Modelling dependence between two random variables
I think you've covered the basics pretty well. The distribution defines all probabilities and moments with regard to relating one with another. correlation only deals with one kind of expectation (i.e. moment) but doesn't imply probabilistic dependence.
In short probability extends to all moments but it's not the reverse (and it's one reason why un-correlatedness does not imply independence but correlation implies dependence). As long as P(A and B) != P(A)P(B) for general events A and B, then there is a dependency between those two events.