= Solution
In <factor analysis>, the independent <multivariate normal distributions> of $Z_i$ and $\xi_i$ imply that their linear combination is also multivariate normal. Its <expected value> is zero and its <covariance matrix> is
$$
\operatorname{Cov}(Y_i)=\Lambda\operatorname{Cov}(Z_i)\Lambda^T+\operatorname{Cov}(\xi_i)
=\Lambda\Lambda^T+\sigma^2I_p.
$$
Therefore
$$
\boxed{Y_i\sim N_p(0,\Lambda\Lambda^T+\sigma^2I_p),\quad i=1,\ldots,n,\text{ independently}.}
$$
The noise variance is assumed positive. The factor contribution is positive semidefinite and has rank at most $k$, while the noise makes the marginal covariance nonsingular.
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