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We have measured the distribution function of the flux decrement D = 1 - e-τ caused by Lyα forest absorption from intervening gas in the lines of sight to high-redshift QSOs from a sample of seven high-resolution QSO spectra obtained with the Keck telescope. The observed flux decrement distribution function (FDDF) is compared with the FDDF from two simulations of the Lyα forest: a ΛCDM model (with Ω = 0.4, Λ = 0.6), computed with the Eulerian code of Cen & Ostriker, and a standard cold dark matter (SCDM) model (with Ω = 1), computed with the smoothed particle hydrodynamics code of Hernquist et al. Good agreement is obtained between the shapes of the simulated and observed FDDFs for both simulations after fitting only one free parameter, which controls the mean flux decrement. The difference between the predicted FDDFs from the two simulations is small, and we show that it arises mostly from a different temperature in the low-density gas (caused by different assumptions that were made about the reionization history in the two simulations), rather than differences between the two cosmological models or numerical effects in the two codes, which use very different computational methods.\nA measurement of the parameter μΩ2b h3/Γ (where Γ is the H I ionization rate due to the ionizing background) is obtained by requiring the mean flux decrement in the simulations to agree with the observed one. Estimating the lower limit Γ > 7 × 10-13 s-1 from the abundance of known QSOs, we derive a lower limit on the baryonic matter density, Ωbh2 > 0.021 (0.017) for the ΛCDM (SCDM) model. The difference between the lower limits inferred from the two models is again due to different temperatures in the low-density gas. We give general analytical arguments for why this lower limit is unlikely to be reduced for any other models of structure formation by gravitational collapse that can explain the observed Lyα forest. When combined with constraints from big bang nucleosynthesis, the large Ωb we infer is inconsistent with some recent D/H determinations (Rugers & Hogan), favoring a low deuterium abundance as reported by Tytler, Fan & Burles. Adopting a fixed Ωb, the measurement of μ(z) allows a determination of the evolution of the ionizing radiation field with redshift. Our models predict an intensity that is approximately constant with redshift, which is in agreement with the assumption that the ionizing background is produced by known quasars for z < 3, but requires additional sources of ionizing photons at higher redshift given the observed rapid decline of the quasar abundance.