Statistical aspects of particle transport in spatially random media in one dimension are studied analytically using the dichotomic and the binomial binary random processes as two models for the spatial randomness in the particle density. The mean and the variance of the flux are expressed in terms of the moments <U(x)n> of the stochastic propagator U(x) = exp[-z(x)], where z(x) is the optical path length. The moments <U(x)n> are rigorously calculable with the above random processes. In the case of semi-infinite media, the calculations are carried out using the one-dimensional, one-speed transport equation. In finite media, one-speed diffusion theory is used to calculate the mean and variance of the flux within the slab. In particular, the statistics of the albedo and the exit current are investigated. The mean of the local reaction rates within the slab is also calculated.