  The exact reaction rates of beta processes for all particles at arbitrary
degeneracy are derived, and an analytic beta-equilibrium condition
$\mu_n=\mu_p+2\mu_e$ for the hot electron-positron plasma with nucleons is
found, if the matter is transparent to neutrinos. This simple analytic formula
is valid only if electrons are nondegenerate, which is generally satisfied in
the hot electron-positron plasma. Therefore, it can be used to efficiently
determine the steady state of the hot matter with positrons.
