In this paper, we establish the expression of photon statistics in the steady state in the micromaser with Kerr medium cavity, study the properties of the atomi c inversion in a stationary state when the atom is in thermal-atom, critical or ultracold-atom regimes in a micromaser. The results show that the properties of the atomic inversion in different regimes are very distinct. In the thermal-atom regime, there are some regions of the cavity length L where the atoms are i nversive, with the increase of the atomic injection rate r, the regions of a tomic inversion and the probability of the atomic inversion increase. In the cri tical regime, atomic inversion exhibits the collapse and revival phenomena. Howe ver, in ultracold-atom regime, there are all regions of the cavity length L where the atoms are not inversive. Kerr effect and the detuning make the probabi lity of the atomic inversion smaller.