  We introduce a method for obtaining the X-ray luminosity function (XLF) and
the binary-period distribution of populations of high-mass X-ray binaries
(HMXBs) in the stellar fields (i.e. outside globular clusters) of normal
galaxies. We start from standard distributions of the parameters of those
primordial binaries which are the progenitors of HMXBs, and follow the
transformation of these ditributions with the aid of a Jacobian formalism as
the former evolve into the latter through the processes of the first mass
transfer and the supernova (SN) that follows. We discuss the distributions of
the post-SN binaries and the HMXBs. We show that our calculated model XLF has a
differential slope $\approx -1.6$ with a flattening at low luminosities, in
excellent agreement with observations. The calculated binary-period
distribution, which basically has a slightly sloping plateau-like character at
intermediate periods, with a rise to this plateau at shorter periods and
fall-off from it at longer periods, is in agreement with the observed
distribution within observational uncertainties. We discuss the physical origin
of these distributions. We demonstrate that, while the effects of both (a) the
distribution of the properties of the massive companion in the HMXBs, and (b)
the primordial orbital distribution and the SN dynamics are important, the
former appear to be dominant in determining the XLF, and the latter in
determining the HMXB binary-period distribution. We discuss the possible roles
of stellar-mass black holes and ultra-luminous X-ray sources (ULX) in the
observed "universal" XLF of HMXBs.
