At temperature $T$, with $\beta=1 /\left(k_{B} T\right)$, the distribution of ultra-relativistic particles with momentum $\mathbf{p}$ is given by

$$n(\mathbf{p})=\frac{1}{e^{\beta p c} \mp 1},$$

where the minus sign is for bosons and the plus $\operatorname{sign}$ for fermions, and with $p=|\mathbf{p}|$.

Show that the total number of fermions, $n_{\mathrm{f}}$, is related to the total number of bosons, $n_{\mathrm{b}}$, by $n_{\mathrm{f}}=\frac{3}{4} n_{\mathrm{b}}$.

Show that the total energy density of fermions, $\rho_{\mathrm{f}}$, is related to the total energy density of bosons, $\rho_{\mathrm{b}}$, by $\rho_{\mathrm{f}}=\frac{7}{8} \rho_{\mathrm{b}}$.