We used the 1.4 GHz NVSS to study radio sources in two color-selected QSO samples: a volume-limited sample of 1313 QSOs defined by M_i < -23 in the redshift range 0.2 < z < 0.45 and a magnitude-limited sample of 2471 QSOs with m_r < 18.5 and 1.8 < z < 2.5. About 10% were detected above the 2.4 mJy NVSS catalog limit and are powered primarily by AGNs. The space density of the low-redshift QSOs evolves as rho proportional to (1+z)^6. In both redshift ranges the flux-density distributions and luminosity functions of QSOs stronger than 2.4 mJy are power laws, with no features to suggest more than one kind of radio source. Extrapolating the power laws to lower luminosities predicts the remaining QSOs should be extremely radio quiet, but they are not. Most were detected statistically on the NVSS images with median peak flux densities S_p(mJy/beam) ~ 0.3 and 0.05 in the low- and high-redshift samples, corresponding to 1.4 GHz spectral luminosities log[L(W/Hz)] ~ 22.7$ and 24.1, respectively. We suggest that the faint radio sources are powered by star formation at rates ~20 M_sun per year in the moderate luminosity (median M_i ~ -23.4) low-redshift QSOs and ~500 M_sun per year in the very luminous (M_i} ~ -27.5) high-redshift QSOs. Such luminous starbursts [ log(L / L_sun) ~ 11.2 and 12.6, respectively] are consistent with quasar mode accretion in which cold gas flows fuel both AGN and starburst.
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