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In this paper, we consider the effects of external gaussian white noise on the full counting statistics (FCS) of non-interacting coherent electrons in mesoscopic systems. We first, derive the wave packet formalism of FCS in the presence of time dependent random potential. We then determine the steady state probability distributions of all the moments and cumulants of particle currents, specially, the shot-noise, the Fano factor, and the distribution in the noise-charge plane by numerically solving the stochastic time dependent Schrödinger equation (STDSE) for one and two-electron systems with transparent boundary conditions, using Numerov extension of Crank-Nicolson method.