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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)

Edit: It appears the more information is needed to answer here.

Assume X is distributed according to the binomial distribution $Bin(n, p)$

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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.

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  • $\begingroup$ I edited the question. $\endgroup$ – Eyal Yeh Feb 22 at 20:47

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