Derive, for all three statistics, the relevant expressions for the quantity \(\left\langle n_{\varepsilon}^{2}ightangle-\left\langle n_{\varepsilon}ightangle^{2}\) from the respective probabilities \(p_{\varepsilon}(n)\). Show that, quite generally,

\[
\left\langle n_{\varepsilon}^{2}ightangle-\left\langle n_{\varepsilon}ightangle^{2}=k T\left(\frac{\partial\left\langle n_{\varepsilon}ightangle}{\partial \mu}ight)_{T}
\]

compare with the corresponding result, (4.5.3), for a system embedded in a grand canonical ensemble.