Derive, for all three statistics, the relevant expressions for the quantity ⟨nε2⟩−⟨nε⟩2 from the respective probabilities pε(n). Show that, quite generally,
⟨nε2⟩−⟨nε⟩2=kT(∂μ∂⟨nε⟩)T,
compare with the corresponding result, (4.5.3), for a system embedded in a grand canonical ensemble.