I want to compute the following expectation value:$$\langle \phi_i ~m_j \rangle$$ Where $\phi_i \sim \text{multinomial}(s_1,(p_1,...,p_N))$ and $m_j\sim\text{multinomial}(s_2,(\alpha_1,...,\alpha_N))$
That is to say that $\phi_i$ is drawn from a multinomial distribution that draws $N$ different colored balls each with probability $p_a$ from a bag of $s_1$ balls. (The probability of drawing type $i$ is $p_i$) Meanwhile $m_j$ is drawn from a multinomial distribution that draws $N$ different numbered balls each with probability $\alpha_a$ from a bag of $s_2$ balls. (The probability of drawing type $j$ is $\alpha_j$)
Now on the surface this problem is easy, $\phi_i$ and $m_j$ appear uncorrelated and thus we can simply say that $$\langle \phi_i~m_j\rangle=\langle\phi_i\rangle\langle m_j\rangle$$
However, $s_1$ and $s_2$ are both random variables drawn from the same distribution specifically $$s_1,s_2\sim \text{multinomial}(T,(S_1,S_2,...,S_M))$$
If we assume that this isn't an issue then we can just say that $$\langle \phi_i~m_j\rangle=\langle\phi_i\rangle\langle m_j\rangle=p_i\langle s_1\rangle\alpha_j\langle s_2\rangle=p_i\alpha_j S_1 S_2 T^2$$
But I'm not sure that this is okay to do. Does the fact that $s_1$ and $s_2$ are drawn from separate distribution that are themselves determined by correlated variables change anything? If so, how?