  We have investigated the basic statistics of the cosmological dispersion
measure (DM) -- such as its mean, variance, probability distribution, angular
power spectrum and correlation function -- using the state-of-the-art
hydrodynamic simulations, IllustrisTNG300, for the fast radio burst (FRB)
cosmology. To model the DM statistics, we first measured the free-electron
abundance and the power spectrum of its spatial fluctuations. The free-electron
power spectrum turns out to be consistent with the dark matter power spectrum
at large scales, but it is strongly damped at small scales ($\lesssim 1$Mpc)
owing to the stellar and active galactic nucleus feedback. The free-electron
power spectrum is well modelled using a scale-dependent bias factor (the ratio
of its fluctuation amplitude to that of the dark matter). We provide analytical
fitting functions for the free-electron abundance and its bias factor. We next
constructed mock sky maps of the DM by performing standard ray-tracing
simulations with the TNG300 data. The DM statistics are calculated analytically
from the fitting functions of the free-electron distribution, which agree well
with the simulation results measured from the mock maps. We have also obtained
the probability distribution of source redshift for a given DM, which helps in
identifying the host galaxies of FRBs from the measured DMs. The angular
two-point correlation function of the DM is described by a simple power law,
$\xi(\theta) \approx 2400 (\theta/{\rm deg})^{-1} \, {\rm pc}^2 \, {\rm
cm}^{-6}$, which we anticipate will be confirmed by future observations when
thousands of FRBs are available.
