= Solution
The design columns are $1$, $x$ and $z$. Their pairwise inner products are zero by the three supplied sums. Put $S_{xx}=\sum_ix_i^2$ and $S_{zz}=\sum_iz_i^2$, positive under the full-rank assumption. Then
$$
X^TX=\operatorname{diag}(n,S_{xx},S_{zz}).
$$
Applying part (a),
$$
\boxed{\widehat\theta=\begin{pmatrix}\widehat\alpha\\\widehat\beta\\\widehat\gamma\end{pmatrix}
=\begin{pmatrix}\overline Y\\\displaystyle\frac{\sum_ix_iY_i}{S_{xx}}\\\displaystyle\frac{\sum_iz_iY_i}{S_{zz}}\end{pmatrix},
\qquad
\operatorname{Var}(\widehat\theta)=\sigma^2\operatorname{diag}\left(\frac1n,\frac1{S_{xx}},\frac1{S_{zz}}\right).}
$$
The off-diagonal covariances vanish because the design columns are <orthogonal>. With Gaussian errors, the coefficient estimators are also jointly normal and therefore independent.
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