  We investigate how the statistical properties of dispersion measure (DM) and
apparent flux density/fluence of (non-repeating) fast radio bursts (FRBs) are
determined by unknown cosmic rate density history [$\rhoFRB (z)$] and
luminosity function (LF) of the transient events. We predict the distributions
of DMs, flux densities, and fluences of FRBs taking account of the variation of
the receiver efficiency within its beam, using analytical models of $\rhoFRB
(z)$ and LF. Comparing the predictions with the observations, we show that the
cumulative distribution of apparent fluences suggests that FRBs originate at
cosmological distances and $\rhoFRB$ increases with redshift resembling cosmic
star formation history (CSFH). We also show that a LF model with a bright-end
cutoff at log$_{10}L_\nu$ [erg s$^{-1}$Hz$^{-1}$] $\sim$ 34 are favored to
reproduce the observed DM distribution if $\rhoFRB (z)\propto$ CSFH, although
the statistical significance of the constraints obtained with the current size
of the observed sample is not high. Finally, we find that the correlation
between DM and flux density of FRBs is potentially a powerful tool to
distinguish whether FRBs are at cosmological distances or in the local universe
more robustly with future observations.
