= Solution
For nondegenerate design let $S_{xx}=\sum_i x_i^2>0$. Maximizing the Gaussian likelihood over the mean parameters is equivalent to minimizing $\sum_i(Y_i-\alpha-\beta x_i)^2$. Since $\sum_ix_i=0$, the <normal equations> separate:
$$
\boxed{\widehat\alpha=\bar Y,\qquad\widehat\beta=\frac{\sum_ix_iY_i}{S_{xx}}.}
$$
Its error is the <linear combination> $\widehat\beta-\beta=\sum_ix_i\epsilon_i/S_{xx}$. Independence and normality therefore give
$$
\boxed{\widehat\beta\sim N(\beta,\sigma^2/S_{xx}).}
$$
If every design point is zero, the slope is not identifiable and no such slope estimator can be inferred; the standard regression answer assumes the stated nondegeneracy.
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