Problem $\mathbf{5}$ (10 pts) Let $X$ and $Y$ be two independent random variables both with mean 0 and variance 1 . Let $Z=X+Y$. Select all statements that are always true. (A) $\mathbb{E}[Z]=0$ (B) $\operatorname{Var}[Z]=1$ (C) $\operatorname{Var}[Z]=2$ (D) For any function $f, \mathbb{E}\left[X^{2} f(Y) ight]=\mathbb{E}[f(Y)]$ (E) For any $a, b, c \in \mathbb{R}, \operatorname{Var}\left[a b X+b^{4} Y+c ight]=a^{2} b^{2}+b^{8}+c^{2}$ (F) It is possible that $H(Z \mid X)=0$ (G) $H(Y \mid X)=H(Y)$ (H) $I(X ; Y)>0$