= Solution
The <Fisher information> is
$$
I(\beta)=\nu\sum_i\mu_i^2x_ix_i^T
=\nu\begin{pmatrix}a&b\\b&c\end{pmatrix}.
$$
Under standard interior, full-rank and large-sample regularity conditions, the <maximum-likelihood estimator> is asymptotically normal with mean $\beta$ and covariance
$$
I(\beta)^{-1}
=\frac1{\nu(ac-b^2)}\begin{pmatrix}c&-b\\-b&a\end{pmatrix}.
$$
Therefore
$$
\boxed{\widehat\beta_2\ \dot\sim\
N\!\left(\beta_2,\frac{a}{\nu(ac-b^2)}\right)}.
$$
The sums already depend on sample size; no extra factor $1/n$ should be inserted. The variance is estimable by substituting fitted means. If all $z_i$ are the same, $ac-b^2=0$ and the two coefficients are not separately identifiable.
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