  The dispersion of fast radio bursts (FRBs) is a measure of the large-scale
electron distribution. It enables measurements of cosmological parameters,
especially of the expansion rate and the cosmic baryon fraction. The number of
events is expected to increase dramatically over the coming years, and of
particular interest are bursts with identified host galaxy and therefore
redshift information.
  In this paper, we explore the covariance matrix of the dispersion measure
(DM) of FRBs induced by the large-scale structure, as bursts from a similar
direction on the sky are correlated by long wavelength modes of the electron
distribution. We derive analytical expressions for the covariance matrix and
examine the impact on parameter estimation from the FRB dispersion measure -
redshift relation. The covariance also contains additional information that is
missed by analysing the events individually. For future samples containing over
$\sim300$ FRBs with host identification over the full sky, the covariance needs
to be taken into account for unbiased inference, and the effect increases
dramatically for smaller patches of the sky. Also forecasts must consider these
effects as they would yield too optimistic parameter constraints. Our procedure
can also be applied to the DM of the afterglow of Gamma Ray Bursts.
