  We develop a sophisticated model of FRB observations, accounting for the
intrinsic cosmological gas distribution and host galaxy contributions, and give
the most detailed account yet of observational biases due to burst width,
dispersion measure, and the exact telescope beamshape. Our results offer a
significant increase in both accuracy and precision beyond those previously
obtained. Using results from ASKAP and Parkes, we present our best-fit FRB
population parameters in a companion paper. Here, we consider in detail the
expected and fitted distributions in redshift, dispersion measure, and
signal-to-noise. We estimate that the unlocalised ASKAP FRBs arise from
$z<0.5$, with between a third and a half within $z<0.1$. Our predicted
source-counts (''logN-logS'') distribution confirms previous indications of a
steepening index near the Parkes detection threshold of $1$ Jy ms. We find no
evidence for a minimum FRB energy, and rule out $E_{\rm min} > 10^{39.0}$ erg
at 90% C.L. Importantly, we find that above a certain DM, observational biases
cause the Macquart (DM-z) relation to become inverted, implying that the
highest-DM events detected in the unlocalised Parkes and ASKAP samples are
unlikely to be the most distant. More localized FRBs will be required to
quantitatively estimate this effect, though its cause is a well-understood
observational bias. Works assuming a 1-1 DM-z relation may therefore derive
erroneous results. Our analysis of errors suggests that limiting factors in our
analysis are understanding of FRB spectral behaviour, sensitivity response of
search experiments, and the treatment of the repeating population and
luminosity function.
