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## Divisibility of cyclic sums

Lately I have been studying the divisibility of some cyclic sums, and I was wondering about the following Conjecture Let it be a set of distinct integers $S=\{x_1,x_2,…,x_n\}$. Then, $$\prod_{n=1}^{n}x_{n}\nmid\left(\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)+\sum_{cyc}\left(\prod_{n=1}^{n-2}x_{n}\right)+…+\sum_{n=1}^nx_n\right)$$ I would appreciate any help regarding this conjecture proof or refutation. I already noted that there exist sets of distinct integers $S=\{x_1,x_2,…,x_n\}$ such that $$\prod_{n=1}^{n}x_{n}\mid\sum_{cyc}\left(\prod_{n=1}^{n-1}x_{n}\right)$$ […]

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## Variance of empirical mean squared error of binomial distribution

Given $m$ binomially distributed samples $X_1,X_2,\ldots,X_m\sim\text{Bin}(n,p)$. Let $\hat{p}_i=X_i/n$ be the corresponding estimates for $p$. The empirical mean squared error is given by $Z= \frac{1}{m}\sum_{i=1}^m (\hat{p}_i-p)^2$. I am looking for the expected value and the variance of $Z$. $E(Z) = p(1-p)/n$ follows directly from the variance of the binomial distribution. Is there a simple way to […]