Covariation of discrete random variables and transformation

Question

Let X and Y be discrete uncorralated identically distributed integer-valued random variables. And $1_{[X=k]} = 1$ if $ X=k$, $ 0 $ otherwise. Can we say something about $ cov(Y, \ 1_{[X=k] }) $ ? I cannot find any counterexample that it should equal zero but I cannot prove that it is zero either.

Answer 1

It is not necessarily $0$; here is a counterexample.

Let $(X,Y)$ be sampled uniformly from $\{(1,2), (2,4), (3,1), (4,3)\}$, so that each of $X$ and $Y$ is uniform on $\{1,2,3,4\}$, and they are uncorrelated despite being as far from independent as possible.

Checking that $\operatorname{Cov}(X,Y)=0$ takes some work, but it's not hard to see that at this point, $\operatorname{Cov}(1_{[X=k]},Y)$ isn't going to be $0$. (It's easiest to get this when $k=2$, since $Y$ attains its highest value of $4$ when $X=2$, and is between $1$ and $3$ when $X \ne 2$.)