# Expected value and modulo

Let $$X \in \mathbb{N}$$ be some discrete RV and define $$Y = X \mod k$$.

The value of $$Y$$ is the representative in the coset of $$X \mod k$$ in $$[0, k-1]$$.

For example if $$X = 9$$ and $$k = 4$$ then $$Y = 1$$.

How can we relate $$E[X]$$ and $$E[Y]$$?

Can we prove that they are close? (depends on some parameters, ofc)

Assume X is distributed according to the binomial distribution $$Bin(n, p)$$
By definition, we have $$E(X)=\sum_{n=1}^\infty P(X=n)\cdot n$$. Given the relationship between $$X$$ and $$Y$$, it follows that $$E(Y)=\sum_{n=0}^{k-1}\left(\sum_{i\equiv n\pmod{k}}P(X=i)\right)\cdot n.$$
Without more information about $$X$$, I'm not sure if you can say anything more precise. I will gladly update if more info is added to the question.