1. Let $X$ and $y$ be two independent random variables with densities $p(x)$ and $p(y) $, respectively. Show the following two properties: $$ \begin{aligned} \mathbb{E}_{p(x, y)}[X+a Y] &=\mathbb{E}_{p(x)} [X]+a \mathbb{E}_{p(y)][Y] 1 \operatorname[Var}_{p(x, y)}[X+a Y] &=\operatorname(Var}_{p(x)}[X]+a^{2} \operatorname[Var}_{p(y)][Y] \end{aligned} $$ for any scalar constant $a \in \mathbb {R} $. Hint: use the definition of expectation and variance, $$ \begin{array}{1} \mathbb{E}_{p(x)}[X]=\int_{x} p(x) x dx W \operatorname{var}_{p(x)} [X]=\mathbb{E}_{p(x)}\left(X^{2} ight]- \mathbb{E}_{p(x)}^{2}[X] \end{array} $$ SE.SD.019