In this paper, we present an analytical approach in obtaining the probability density function (pdf) of the random decision variable Y, formed at the output of power-cubic all-optical nonlinear preprocessor followed by the photodetector. Our approach can be used to accurately evaluate the performance of ultrafast pulse detection in the presence of Gaussian noise. Through rigorous Monte-Carlo simulation, the accuracy of widely used Gaussian approximation of decision variable Y is refuted. However, in this paper we show that the so called Log-Pearson type-3 probability density function (LP3 pdf) is an excellent representation for the decision variable Y . Three distinguishable parameters of the LP3 pdf are obtained through analytical derivation of three moments of the decision variable Y . Furthermore, toward a more realistic model, in addition to ASE Gaussian noise, the effects of shot and thermal noises are also included. Finally, using the presented analytical approach, it is shown that power-cubic preprocessor outperforms its quadratic counterparts, i.e., Second Harmonic Generation (SHG) and Two Photon Absorption (TPA) devices, in high power regime where shot and thermal noises can be neglected.