  In this paper, we investigate the effects of random fluctuations of the Earth
matter density for long baselines on the neutrino oscillation transition
probabilities. We especially identify relevant parameters characterizing the
matter density noise and calculate their effects by averaging over statistical
ensembles of a large number of matter density profiles. For energies and
baselines appropriate to neutrino factories, absolute errors on the relevant
appearance probabilities are at the level of $|\Delta P_{\alpha \beta}| \sim
10^{-4}$ (with perhaps $|\Delta P_{\mu e}|/P_{\mu e} \sim 1%$ for neutrinos),
whereby a modest improvement in understanding of the geophysical data should
render such effects unimportant.
