= Solution
The <ordinary least squares> estimator is $\widehat\beta=(X^TX)^{-1}X^TY$. Consequently the <fitted values> and <regression residuals> are
$$
\widehat Y=X\widehat\beta=PY,\qquad e=Y-\widehat Y=(I_n-P)Y.
$$
An affine transformation of a <multivariate normal distribution> is again <multivariate normal>, possibly with a singular <covariance matrix>. For a <random vector> with <covariance matrix> $\Sigma$, its transformed <covariance matrix> is $A\Sigma A^T$. Since $PX=X$, $P^2=P=P^T$ and $(I-P)^2=I-P$, these results give
$$
\boxed{\widehat Y\sim N_n(X\beta,\sigma^2P),\qquad e\sim N_n(0,\sigma^2(I_n-P)).}
$$
Both <multivariate normal distributions> are supported on their respective projected subspaces. In particular, $\widehat Y_i$ has <variance> $\sigma^2P_{ii}$ and $e_i$ has <variance> $\sigma^2(1-P_{ii})$; the <regression residuals> need not be mutually independent.
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