Statistical moments of laser intensity passing through an amplifier with stochastic gain are evaluated. This situation happens when turbulent flows exist in the gain medium, which is assumed to be a two-level atomic system. Also, it is assumed that the turbulence structure is fully characterized by a stochastic process. Meanwhile, this gain medium is assumed to be partially-homogeneously broadened and slightly saturated. In our derivations, the paraxial approximation is combined with the perturbation method to solve for the random laser field inside the amplifier. For mathematical simplicity, a plane wave is propagated through the gain medium which contains only weak turbulence fluctuations. Dependences of the average laser intensity and contrast on turbulence fluctuation strength, characteristic length of the turbulence, and gain saturation are numerically demonstrated. It is found that the turbulence fluctuations make both average intensity and contrast stronger. Also, the gain saturation reduces both average intensity and contrast. In addition, it is shown that the contrast increases as the characteristic length of turbulence is increased. This trend is different from that in the random wave theory when there is no gain or loss. In the conventional theory of random wave propagation without gain or loss, the contrast is usually decreasing as the characteristic length increases.