  In a seminal paper, Page found the exact formula for the average entanglement
entropy for a pure random state. We consider the analogous problem for the
ensemble of pure fermionic Gaussian states, which plays a crucial role in the
context of random free Hamiltonians. Using recent results from random matrix
theory, we show that the average entanglement entropy of pure random fermionic
Gaussian states in a subsystem of $N_A$ out of $N$ degrees of freedom is given
by $\langle
S_A\rangle_\mathrm{G}\!=\!(N\!-\!\tfrac{1}{2})\Psi(2N)\!+\!(\tfrac{1}{4}\!-\!N_A)\Psi(N)\!+\!(\tfrac{1}{2}\!+\!N_A\!-\!N)\Psi(2N\!-\!2N_A)\!-\!\tfrac{1}{4}\Psi(N\!-\!N_A)\!-\!N_A$,
where $\Psi$ is the digamma function. Its asymptotic behavior in the
thermodynamic limit is given by $\langle S_A\rangle_\mathrm{G}\!=\! N(\log
2-1)f+N(f-1)\log(1-f)+\tfrac{1}{2}f+\tfrac{1}{4}\log{(1-f)}\,+\,O(1/N)$, where
$f=N_A/N$. Remarkably, its leading order agrees with the average over
eigenstates of random quadratic Hamiltonians with number conservation, as found
by Lydzba, Rigol and Vidmar. Finally, we compute the variance in the
thermodynamic limit, given by the constant $\lim_{N\to\infty}(\Delta
S_A)^2_{\mathrm{G}}=\frac{1}{2}(f+f^2+\log(1-f))$.
