Exact variance of Pearson's chi-square statistic
Combinatorics · proposed by @Ale712
A fair six-sided die is rolled independently $10$ times. Let $N_i$ denote the number of times face $i$ appears, for $i=1,\ldots,6$.
Consider Pearson's chi-square goodness-of-fit statistics
$$
Q=\sum_{i=1}^{6}\frac{(N_{i}-10/6)^2}{10/6}.
$$
What is the exact value of $\operatorname{Var}(Q)$?