[Math] Distribution of a product of Multinomials

Question

Consider the following: $(X_1, X_2, X_3, X_4) \sim \mathrm{Multinomial} (n,\mathbf{p})$ where $\mathbf{p} = (p_1,p_2,p_3,p_4)$. I would like to find the distribution of $X_1 X_4$, or at least know some bounds on the variance of $X_1 X_4$. I know that $E(X_1X_4) = n^2 p_1 p_4 – np_1 p_4 = n(np_1p_4 – p_1p_4)$. Is $X_1X_4$ in any way

Answer 1

$$Var(X_1X_4)=\mathbb{E}[X_1X_4]^2-\left(\mathbb{E}[X_1X_4]\right)^2$$

and

$$\mathbb{E}[X_1X_4]=p_1p_4\times n(n-1)$$

$$\mathbb{E}[X_1X_4]^2=p_1p_4\times n\left[(n-1)+(p_1+p_4)(n^2-3n+2)+p_1p_4(n^3-6n^2+11n-6)\right]$$