On the normalized FRB luminosity function

33 fast radio bursts (FRBs) had been detected by 2018 March. Although the sample size is still limited, meaningful statistical studies can already be carried out. The normalized luminosity function places important constraints on the intrinsic power output, sheds light on the origin(s) of FRBs, and can guide future observations. In this paper, we measure the normalized luminosity function of FRBs. Using Bayesian statistics, we can naturally account for a variety of factors such as receiver noise temperature, bandwidth, and source selection criteria. We can also include astronomical systematics, such as host galaxy dispersion measure, FRB local dispersion measure, galaxy evolution, geometric projection effects, and Galactic halo contribution. Assuming a Schechter luminosity function, we show that the isotropic luminosities of FRBs have a power-law distribution that covers approximately three orders of magnitude, with a power-law index ranging from -1.8 to -1.2 and a cut off {∼ } 2× 10^{44} erg s^{-1}. By using different galaxy models and well-established Bayesian marginalization techniques, we show that our conclusions are robust against unknowns, such as the electron densities in the Milky Way halo and the FRB environment, host galaxy morphology, and telescope beam response.