If we choose the quantities \(E\) and \(V\) as "independent" variables, then the probability distribution function (15.1.8) does not reduce to a form as simple as (15.1.13) or (15.1.15); it is marked instead by the presence, in the exponent, of a cross term proportional to the product \(\triangle E \Delta V\).

Consequently, the variables \(E\) and \(V\) are not statistically independent \(: \overline{(\Delta E \Delta V)} eq 0\).

Show that

\[
\overline{(\Delta E \Delta V)}=k T\left\{T\left(\frac{\partial V}{\partial T}\right)_{P}+P\left(\frac{\partial V}{\partial P}\right)_{T}\right\}
\]

verify as well expressions (15.1.14) and (15.1.18) for \(\overline{(\Delta V)^{2}}\) and \(\overline{(\Delta E)^{2}}\).

[Note that in the case of a two-dimensional normal distribution, namely

\[
p(x, y) \propto \exp \left\{-\frac{1}{2}\left(a x^{2}+2 b x y+c y^{2}\right)\right\}
\]

the quantities \(\left\langle x^{2}\rightangle,\langle x yangle\), and \(\left\langle y^{2}\rightangle\) can be obtained in a straightforward manner by carrying out a logarithmic differentiation of the formula

\[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp \left\{-\frac{1}{2}\left(a x^{2}+2 b x y+c y^{2}\right\} d x d y=\frac{2 \pi}{\sqrt{\left(a c-b^{2}\right)}}\right.
\]

with respect to the parameters \(a, b\), and \(c\). This leads to the covariance matrix of the distribution, namely

\[
\left(\begin{array}{ll}
\left\langle x^{2}\rightangle & \langle x yangle \\
\langle y xangle & \left\langle y^{2}\rightangle
\end{array}\right)=\frac{1}{\left(a c-b^{2}\right)}\left(\begin{array}{rr}
c & -b \\
-b & a
\end{array}\right)
\]

If \(b=0\), then

\[
\left.\left\langle x^{2}\rightangle=1 / a, \quad\langle x yangle=0, \quad\left\langle y^{2}\rightangle=1 / c .\right]^{22}
\]