# Covariance of square root for two bins of a multinomial

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

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

• What's wrong with the question ? – ippiki-ookami Mar 10 at 7:59
• 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. – cdipaolo Mar 10 at 8:10

## 1 Answer

If you are okay with bounds, the function $$x \mapsto \sqrt{x}$$ is concave and thus Jensen's inequality tells us $$0 \leq \mathbb{E} \sqrt{X_i X_j}\leq \sqrt{\mathbb{E} X_i X_j} = \sqrt{(n^2-n)p_ip_j} \leq n\sqrt{p_ip_j}.$$

• 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. – ippiki-ookami Mar 10 at 13:09