Bohr exactly predictedsaid that an electron would revolve in certain stationary orbits and he gave its mathematical interpretation.
While in quantum mechanics we deal with probability of finding an electron around the nucleus so we can't derive the Bohr's postulates as Bohr said that "electrons revolve" in circular orbits while QED says "electrons may be found" in region around nucleus but Bohr's theory is quite close to what predicted by QED.
So,for ground state the linear probability density $P(r)$ is given as $$ P(r)=\frac {4r^2e^{\frac{-2r}{a_0}}}{a_0^3}\ . $$ The wave function of electron in ground state is also proportional to $e^{\frac{-2r}{a_0}}$. So it is clear from equations that the radius predicted by Bohr's model is quite close to thatthe probability region predicted by QED.