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Relative intensity noise (RIN) is one of the most significant artifacts that ultimately limits the use of broadband Raman amplifier applications in optical communications systems. Recently, as a way to alleviate some of the detrimental features of continuously pumped Raman amplifiers, pulsed pumping was introduced. In this paper, we study the RIN transfer in pulse-pumped fiber Raman amplifiers. An analytical expression describing RIN transfer from a pump with an arbitrary pulse shape has been derived for co- and counter-pumping configurations. The dependence of RIN transfer on pump modulation frequency, depth of modulation, and duty cycle in a typical 80-km fiber span has been analyzed for pumps with sinusoidal and rectangular profiles. We show that in the case of rectangular pulsing of pump, RIN transfer in counter-pumping regime is substantially enhanced compared with that of continuous wave (CW) pumping due to the presence of resonant peaks at the pump intensity harmonics. In the co-propagation regime, RIN transfer for pulsed pumping does not exhibit resonant peaks and is generally lower than that for CW pumping. These features are reflected on the Q-penalty and may have detrimental effects on the system performance.