Simplification Formula (conditional expec)

Question

Is there any known formula linking $E[D\mid (S=0) \cap (X=j)]$ and $E[D\mid (S=0)]$ given that S and X are 2 indepedent random variables?

Thanks

Answer 1

The answer is no, but I think you might be meaning to give more information.

There are a bunch of scenarios where different things could happen. For instance, suppose that D,S, and X are all independent, then we have that

$$E(D \mid (S=0) \cap (X=j)) = E(D)$$

As another case, if D is independent of S but not X, then we have that

$$E(D \mid S=0) = E(D)$$

but we don't know what $E(D \mid (S=0) \cap (X = j))$ is.

The issue is that, since you're taking the expectation with respect to $D$, you have to know the dependence structure of all three random variables, rather than just the two you are conditioning on, to potentially link the expectations.