= Solution
A grand <canonical ensemble> describes a system exchanging both energy and particles with a reservoir at fixed $T$, volume, and <chemical potential> $\mu$. For a noninteracting fermion level $r$, occupation is $0$ or $1$, so
$$
\mathcal Z_r=1+e^{-\beta(E_r-\mu)}.
$$
Factorization over levels gives $\mathcal Z=\prod_r\mathcal Z_r$, and <differentiation> with respect to $\beta\mu$ yields the <Fermi-Dirac distribution>
$$
\boxed{\langle n_r\rangle=\frac{1}{e^{\beta(E_r-\mu)}+1}.}
$$
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