Making use of expressions (15.1.11) and (15.1.12) for \(\Delta S\) and \(\Delta P\), and expressions (15.1.14) for \(\overline{(\Delta T)^{2}}, \overline{(\Delta V)^{2}}\), and \(\overline{(\Delta T \Delta V)}\), show that
(a) \(\overline{(\Delta T \Delta S)}=k T\);
(b) \(\overline{(\Delta P \Delta V)}=-k T\);
(c) \(\overline{(\Delta S \Delta V)}=k T(\partial V / \partial T)_{P}\);
(d) \(\overline{(\Delta P \Delta T)}=k T^{2} C_{V}^{-1}(\partial P / \partial T)_{V}\).

Note that results (a) and (b) give: \(\overline{(\Delta T \Delta S-\Delta P \Delta V)}=2 k T\), which follows directly from the probability distribution function (15.1.8).