The evolution of randomly modulated solitons in the KdV equation is investigated. The cases of multiplicative and additive noises are considered. The distribution function for the soliton parameters is found using the inverse scattering transform. It is shown that the distribution function has non-Gaussian form and that the most probable and the mean value of the soliton amplitudes are distinct. The analytical results agrees well with the results of the numerical simulations of the KdV equation with random initial conditions. The results obtained for the KdV equation is used to discuss the evolution of randomly modulated small-amplitude dark solitons in optical fibers and pulses in a nonlinear transmission lines.