We report experimental observation and characterization of rogue wave-like extreme value statistics arising from pump-signal noise transfer in a fiber Raman amplifier. Specifically, by exploiting Raman amplification with an incoherent pump, the amplified signal is shown to develop a series of temporal intensity spikes whose peak power follows a power-law probability distribution. The results are interpreted using a numerical model of the Raman gain process using coupled nonlinear Schrödinger equations, and the numerical model predicts results in good agreement with experiment.