This paper presents results of experimental and theoretical studies of light transmission through optical fibers with disorder generated in its germanium-doped core via UV radiation transmitted through a diffuser. The experimental results on transmission of the radiation of 543 nm wavelength demonstrate the presence of the disorder in the core of the optical fiber - beyond a certain characteristic length, the transmitted power is observed to be distributed over all modes of the fiber. A theoretical model based on coupled mode theory is developed. An analytical expression for the mixing length is obtained and agrees well with the experiment. For long sections of disordered fiber, the experimentally measured distribution of the near-field intensity at the output surface of the fiber is well described by the Rayleigh negative exponential function. This suggests a statistically uniform distribution of the transmitted power over all modes, that agrees with the prediction of the theoretical model. The reported technique provides an easy way to fabricate different configurations of controlled disorder in optical fibers suitable for such applications as random fiber lasers.