We investigate quantum fluctuations in optical bistability upon nonclassical pumping by a sub-Poissonian laser. The bistable system consists of an optical cavity filled with a resonant atomic medium. We evaluate analytically the photocurrent noise spectrum at the output of the bistable system and study its behavior as a function of different control parameters. We discover that for absorptive bistability, the regular photon statistics of the pump field is destroyed at the output of the bistable system. On the contrary, for the dispersive bistability, we find complete noise reduction at the output of the system close to the upper turning point on the bistability curve. We investigate the sensitivity of this noiseless point and find it to be reasonably rigid.