Assume that $X_1$, $X_2$ and $Y$ are normally distributed, and are correlated with $\rho_1$ and $\rho_2$, respectively. $X$ has two states in the sense that its distribution is $X=X_1$ with probability $q$, and $X=X_2$ w.p. $1-q$. Suppose that we know the followings: $$E[Y \mid X_i=x], \quad \text{and} \quad V[Y \mid X_i=x] \quad for \quad i \in \{ 1,2 \}. $$ The question is can we have $V[Y \mid X=x]$ in terms of above known forms?
I attempted by using conditional variance, but I am lost at some point. More specifically, when I do so, I need to find $E[Y^2 \mid X=x],$ which is the point I am stuck.