= Solution
Because the scalar signal and the noise vector are <independent random variables> with <normal distributions>, their sum is a <Gaussian random vector>. Its <expected value> is zero, and the <covariance matrix> is
$$
\boxed{\Sigma_S=I_d+\theta u(S)u(S)^\top,\qquad P_{\theta,S}=N(0,\Sigma_S).}
$$
Thus this is a <rank-one covariance spike>. Each summand $X_iX_i^\top$ has <expected value> $\Sigma_S$, so \b[$\mathbb E\widehat\Sigma=\Sigma_S$]. Here the empirical matrix is an <uncentered empirical second-moment matrix>; because the population <expected value> is known to be zero, it is an <unbiased estimator> of the <covariance matrix>. The TeX's apparent factorial denominator is a transcription error.
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