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The covariance statistics of measurements by a fully polarimetric microwave radiometer are derived using a fundamental noise-theoretic approach. Previous published derivations of a similar nature have only included the third Stokes brightness temperature. The results are confirmed by a series of numerical Monte Carlo simulations of the underlying radiometric measurement process. It is found that the additive noise that is present in the measurements can be correlated between polarimetric channels and that the correlation statistics will vary as a function of the polarization state of the scene under observation. General expressions are also derived for the measurement precision (the radiometric NEΔT) and the system noise temperature of the third and fourth Stokes channels. It is found that both the precision and noise temperature can also depend on the polarization state of the scene.