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Take $(X_1, \dots, X_k) \sim Multinomial(n, (p_1, \dots, p_k))$.

Do we have a closed form expression for $\mathbb{E}[\sqrt{X_i X_j}], i\neq j$ ?

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  • $\begingroup$ What's wrong with the question ? $\endgroup$ – ippiki-ookami Mar 10 at 7:59
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    $\begingroup$ I personally think it is a good question, but sometimes people want askers to include more details about what they have tried. Perhaps that's what the down-voter had an issue with. $\endgroup$ – cdipaolo Mar 10 at 8:10
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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 $$0 \leq \mathbb{E} \sqrt{X_i X_j}\leq \sqrt{\mathbb{E} X_i X_j} = \sqrt{(n^2-n)p_ip_j} \leq n\sqrt{p_ip_j}.$$

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    $\begingroup$ Thank you cdipaolo. Yes, notice that this bound can also be obtained with Cauchy-Schwartz inequality $\mathbb{E}[\sqrt{X_i X_j}] \leq \sqrt{\mathbb{E}[X_i]\mathbb{E}[X_j]} \leq n \sqrt{p_i p_j}$. This is not sufficient for my problem at hand unfortunately. $\endgroup$ – ippiki-ookami Mar 10 at 13:09

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