  We present a systematic study of the X-ray binaries (XRBs) containing a black
hole (BH) and a non-degenerate companion, in which mass transfer takes place
via either capturing the companion's wind or Roche lobe overflow (RLO). As
shown in our previous work that focusing on the formation and evolution of
detached BH binaries, our assumed models relevant to BH's progenitors predicted
significantly different binary properties \citep{sl19}. In this paper, we
further follow the evolutionary paths of the BH systems that appearing as XRBs.
By use of both binary population synthesis and detailed binary evolution
calculations, we can obtain the potential population of BH XRBs. Distributions
at the current epoch of various binary parameters have been computed. The
observed sample of wind-fed XRBs can be well reproduced under assumption of all
our models. The wind-fed XRBs are expected to be so rare ($ \lesssim 100 $)
that only a couple of such systems have been detected. Comparison of known RLO
XRBs with the calculated distributions of various binary parameters indicates
that only the models assuming relatively small masses for BH progenitors can
roughly match the observations. Accordingly we estimate that there are hundreds
of RLO XRBs in the Milky Way, of which the majority are low-mass XRBs. The RLO
systems may become ultraluminous X-ray sources (ULXs) if the BH accretes at a
very high rate, we expect that about a dozen ULXs with a BH accretor may exist
in a Milky Way$-$like galaxy.
