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A statistical model for the polarization of pulsar radio emission is enhanced to account for the heavy modulation of the emission, the possible covariance of the Stokes parameters, and the observed asymmetries in the distributions of total intensity, polarization, and fractional polarization by treating the intensities of the orthogonal polarization modes as exponential random variables. The model is used to derive theoretical distributions to compare with what is observed. The resulting distributions are unimodal and generally asymmetric. The unimodality arises from the model's fundamental assumption that the orthogonal modes are superposed. The asymmetry originates primarily from different fluctuations in mode intensities. The distributions of fractional polarization are truncated at the degree of linear and circular polarization intrinsic to the modes. A number of observable parameters that quantify the statistical properties of the emission and its polarization are derived and are shown to be functions only of the ratio of the modes' mean intensities, M, suggesting their spectra coevolve according to the frequency dependence of M. This particular implementation of the model requires the modes to fluctuate differently in order to replicate the observations. Since a single underlying emission mechanism seems unlikely to selectively modulate the mode intensities, the different fluctuations are attributed either to different emission mechanisms for the modes or to mode-dependent propagation or scattering effects in the pulsar magnetosphere.