\begin{question}{1.6}
	Show that if two varaibles $x$ and $y$ are independent, then their covariance is zero.
\end{question}

\begin{answer}{}
	The covariance of $x$ and $y$ is given by
	\begin{align}
		\cov{x, y} &= \mathbb{E}_{x,y}\left[\left\{x - \ev{x}\right\} \left\{y - \ev{y}\right\} \right]\\
		&= \mathbb{E}_{x,y}\left[xy\right] - \ev{x}\ev{y}.
	\end{align} 
	When two variables $x$ and $y$ are independent, then their joint distribution $p(x,y)$ satisfies $p(x,y) = p(x)p(y)$. It can be seen that
	\begin{align}
		\mathbb{E}_{x,y}\left[xy\right] &= \int\int p(x,y)xy\ dxdy\\
		&= \int\int p(x)p(y)xy\ dxdy\\
		&= \int p(x)x \ dx \int p(y)y\ dy\\
		&= \ev{x}\ev{y}.
	\end{align}
	Therefore we can conclude that if $x$ and $y$ are independent, then $\cov{x,y} = 0$.
\end{answer}