Ultra-luminous X-ray Sources in nearby galaxies from ROSAT HRI   observations II. statistical properties
  The statistical properties of the ultra-luminous X-ray source (ULX) populations extracted from the ROSAT HRI survey of X-ray point sources in nearby galaxies in Paper I are studied to reveal connections between the ULX phenomenon and survey galaxy properties. The $logN$--$logS$ relation is used to calculate and remove the influence of false ULXs from the background and/or foreground objects. Study of the luminosity functions shows that the regular ULXs below $10^{40}$ erg/sec are an extension of the ordinary HMXB/LMXB population below $10^{39}$ erg/sec in the late-type galaxies, and that the extreme ULXs above $10^{40}$ erg/sec might be a different population from the regular ULXs. This survey confirms statistically that the ULX phenomenon is closely connected to star formation activities, since ULXs preferentially occur in late-type galaxies rather than in early-typTwo ULX populations, the HMXB-like ULXs as an extension of the ordinary HMXB population associated with the young stellar population and the LMXB-like ULXs as an extension of the ordinary LMXB population associated with the old stellar population, are both required to account for the total ULX population. It is found that the LMXB-like ULXs dominate the ULX population at low star formation rates, while HMXB-like ULXs dominate at high star formation rates. However, an accurate quantitative description of the relative fractions of HMXB-like ULXs and LMXB-like ULXs is impossible in the current HRI survey due to the source blending effects, and a Chandra survey of the X-ray point sources in nearby galaxies would be required for this purpose. e galaxies, and ULXs in late-type galaxies tend to trace the spiral arms. 
Ji-Feng Liu, Joel N. Bregman, and Jimmy Irwin