  Fast radio bursts (FRBs) are millisecond-duration radio transients from
cosmological distances. Their isotropic energies follow a power-law
distribution with a possible exponential cutoff, but their intrinsic redshift
distribution, which contains information about the FRB sources, is not well
understood. We attempt to constrain both distributions by means of Monte Carlo
simulations and comparing the simulations results with the available FRB
specific fluence distribution, dispersion measure (DM) distribution, and the
estimated energy distribution data. Two redshift distribution models, one
tracking the star formation history of the universe and another tracking
compact binary mergers, are tested. For the latter model, we consider three
merger delay timescale distribution (Gaussian, log-normal, and power law)
models. Two FRB samples detected by Parkes and the Australian Square Kilometre
Array Pathfinder (ASKAP), respectively, are used to confront the simulation
results. We confirm the $\sim -1.8$ power law index for the energy distribution
but the exponential cutoff energy of the distribution, if any, is
unconstrained. For the best energy distribution model, none of the redshift
distributions we considered are rejected by the data. A future, larger, uniform
FRB sample (such as the one collected by the Canadian Hydrogen Intensity
Mapping Experiment, CHIME) can provide better constraints on the intrinsic FRB
redshift distribution using the methodology presented in this paper.
