Prove that energy fluctuations in a canonical distribution are given by

$$\left\langle(E-\langle E\rangle)^{2}\right\rangle=k_{B} T^{2} C_{V}$$

where $T$ is the absolute temperature, $C_{V}=\left.\frac{\partial\langle E\rangle}{\partial T}\right|_{V}$ is the heat capacity at constant volume, and $k_{B}$ is Boltzmann's constant.

Prove the following relation in a similar manner:

$$\left\langle(E-\langle E\rangle)^{3}\right\rangle=k_{B}^{2}\left[\left.T^{4} \frac{\partial C_{V}}{\partial T}\right|_{V}+2 T^{3} C_{V}\right]$$

Show that, for an ideal gas of $N$ monatomic molecules where $\langle E\rangle=\frac{3}{2} N k_{B} T$, these equations can be reduced to

$$\frac{1}{\langle E\rangle^{2}}\left\langle(E-\langle E\rangle)^{2}\right\rangle=\frac{2}{3 N} \quad \text { and } \quad \frac{1}{\langle E\rangle^{3}}\left\langle(E-\langle E\rangle)^{3}\right\rangle=\frac{8}{9 N^{2}}$$