 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

Here $\widehat\alpha=\alpha+n^{-1}\sum_i\epsilon_i$. By <linearity of covariance> and <independence>,
$$
\boxed{\operatorname{Cov}(\widehat\alpha,\widehat\beta)
=\frac1{nS_{xx}}\sum_{i,j}x_j\operatorname{Cov}(\epsilon_i,\epsilon_j)
=\frac{\sigma^2}{nS_{xx}}\sum_i x_i=0.}
$$
Thus the two <estimators> are <uncorrelated random variables>.

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