Spread of a multinomial count gap

HardRandom Variables Statistics~7m

Roll a weighted die $n = 10$ times independently. Let $N_i$ count how many rolls show face $i$ (probability $p_i = 0.2$ ) and $N_j$ count face $j$ (probability $p_j = 0.3$ ). What is $\operatorname{Var}(N_i - N_j)$ , the variance of the gap between the two counts?

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Var(N_i - N_j) = Var(N_i) + Var(N_j) - 2 Cov(N_i, N_j), with the multinomial covariance Cov = -n p_i p_j.
The covariance is negative, so -2 Cov adds a positive term: 1.6 + 2.1 + 1.2.

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4.9 (± 0.01)

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Asked at: Jane Street, Two Sigma

Spread of a multinomial count gap

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