Solution
= Solution
The density is that of a <Gamma distribution> with shape $\nu>0$ and rate $\nu/\mu_i$, requiring $\mu_i>0$. Direct integration gives
$$
\mathbb E(Y_i^r)
=\left(\frac{\mu_i}{\nu}\right)^r\frac{\Gamma(\nu+r)}{\Gamma(\nu)}
$$
for $\nu+r>0$. Using $\Gamma(\nu+1)=\nu\Gamma(\nu)$ and $\Gamma(\nu+2)=\nu(\nu+1)\Gamma(\nu)$,
$$
\boxed{\mathbb E Y_i=\mu_i,\qquad
\operatorname{var}Y_i=\mu_i^2/\nu}.
$$