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We derive an analytical theory for the noise figure of an undepleted and lossless fiber optical parametric amplifier (FOPA). Both the signal and the wavelength converted idler are investigated. Our theory is applicable for both an ideal pump power source, as well as a noisy one. We find that a noisy pump source can severely degrade the performance at high gain due to the stochastic gain-variations the signal and idler will experience. The theory is compared with Monte Carlo simulations of the FOPA and an excellent agreement is obtained. Simulations in the gain-depleted region show the possibility to reach below quantum-limited, phase-insensitive amplification for single channel transmission.