Propagation of incoherent nonlinear picosecond pulses in a single-mode optical fiber is considered within the framework of the nonlinear Schrödinger equation. Statistical characteristics of a non-soliton wave packet induced by a random input pulse are obtained with the help of the inverse scattering technique. The probability of a soliton creation from the random pulse is found. Mean soliton amplitude and velocity changings caused by small fluctuations of the soliton-like input pulse are calculated. The influence of dissipative losses on the obtained results is discussed.