Suppose $X_1,X_2$ are random variables. Discrete. Also, assume $Y_1 =F_1(X_1), Y_2=F_2(X_2)$. Prove the following relation.
$$I(X_1;X_2) \ge I(Y_1;Y_2)$$
I think the solution like this. Is it right?
Given $F_2$, since $X_1 \rightarrow X_2 \rightarrow F_2(X_2) $, \begin{equation} I(X_1,X_2) \ge I(X_1,F_2(X_2)) \end{equation} also Given $F_1$, Since $F_2(X_2) \rightarrow X_1 \rightarrow F_1(X_1) $ $$I(X_1,F_2(X_2)) \ge I(F_1(X_1),F_2(X_2))$$