The interplay of such cornerstones of modern nonlinear fiber optics as a nonlinearity, stochasticity and polarizationleads to variety of the noise induced instabilities including polarizationattraction and escape phenomena harnessing of which is a key to unlocking thefiber optic systems specifications required in high resolution spectroscopy,metrology, biomedicine and telecommunications. Here, by using direct stochasticmodeling, the mapping of interplay of the Raman scattering-based nonlinearity,the random birefringence of a fiber, and the pump-to-signal intensity noisetransfer has been done in terms of the fiber Raman amplifier parameters, namelypolarization mode dispersion, the relative intensity noise of the pump laser,fiber length, and the signal power. The obtained results reveal conditions foremergence of the random birefringence-induced resonance-like enhancement of thegain fluctuations (stochastic anti-resonance) accompanied by pulse broadeningand rare events in the form of low power output signals having probabilityheavily deviated from the Gaussian distribution.