\item \subquestionpoints{5}
Derive an expression for the mean of the distribution. Show that
$\mathbb{E}[Y; \eta] = \frac{\partial}{\partial\eta}a(\eta)$ (note that
$\mathbb{E}[Y; \eta] = \mathbb{E}[Y | X; \theta]$ since $\eta = \theta^T x$).
In other words, show that the mean of an exponential family distribution is the
first derivative of the log-partition function with respect to the natural
parameter.

\textbf{Hint:} Start with observing that $\frac{\partial}{\partial \eta} \int
p(y;\eta) dy = \int \frac{\partial}{\partial \eta} p(y;\eta) dy$.