  In this paper, we investigate the FRB population using the first CHIME/FRB
catalog. We first reconstruct the extragalactic dispersion measure -- redshift
relation ($\mathrm{DM_E} - z$ relation) from well-localized FRBs, then use it
to infer redshift and isotropic energy of the first CHIME/FRB catalog. The
intrinsic energy distribution is modeled by the power law with an exponential
cutoff, and the selection effect of the CHIME telescope is modeled by a
two-parametric function of specific fluence. For the intrinsic redshift
distribution, the star formation history (SFH) model, as well as other five
SFH-related models are considered. We construct the joint likelihood of
fluence, energy and redshift, and all the free parameters are constrained
simultaneously using Bayesian inference method. The Bayesian information
criterion (BIC) is used to choose the model that best matches the observational
data. For comparison, we fit our models with two data samples, i.e. the Full
sample and the Gold sample. The power-law index and cutoff energy are tightly
constrained to be $1.8 \lesssim \alpha \lesssim 1.9$ and $\mathrm{log}(E_c/{\rm
erg}) \approx 42$, which are almost independent of the redshift distribution
model and the data sample we choose. The parameters involving the selection
effect strongly depends on the data sample, but are insensitive to the redshift
distribution model. According to BIC, the pure SFH model is strongly disfavored
by both the Full sample and Gold sample. For the rest five SFH-related redshift
distribution models, most of them can match the data well if the parameters are
properly chosen. Therefore, with the present data, it is still premature to
draw a conclusive conclusion on the FRB population.
