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Bounding Detection and Estimation by SNR in Gaussian Channel

Going directly for the variance is hard, because we don't know the pmf/pdf/cdf/moment-generating function/characterstic function/… of your $X$. What we can do, instead, is going through entropy, and ...
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In this question, what does μ stand for?

Hint: Under hypothesis $\mathcal H_1$ when $S = 1$, $\mathbf X = [\mu_1, \mu_2, \cdots, \mu_N] + [n_1, n_2, \cdots, n_N]$ while under hypothesis $\mathcal H_0$ when $S = 0$, $\mathbf X = [0, 0, \cdots,...
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