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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.