Take $(X_1, \dots, X_k) \sim Multinomial(n, (p_1, \dots, p_k))$.
Do we have a closed form expression for $\mathbb{E}[\sqrt{X_i X_j}], i\neq j$ ?
2026-03-25 21:48:58.1774475338
Covariance of square root for two bins of a multinomial
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If you are okay with bounds, the function $x \mapsto \sqrt{x}$ is concave and thus Jensen's inequality tells us $$0 \leq \mathbb{E} \sqrt{X_i X_j}\leq \sqrt{\mathbb{E} X_i X_j} = \sqrt{(n^2-n)p_ip_j} \leq n\sqrt{p_ip_j}.$$