Probability distribution function for inclinations of merging compact
  binaries detected by gravitational wave interferometers
  We analytically discuss probability distribution function (PDF) for
inclinations of merging compact binaries whose gravitational waves are
coherently detected by a network of ground based interferometers. The PDF would
be useful for studying prospects of (1) simultaneously detecting
electromagnetic signals (such as gamma-ray-bursts) associated with binary
mergers and (2) statistically constraining the related theoretical models from
the actual observational data of multi-messenger astronomy. Our approach is
similar to Schutz (2011), but we explicitly include the dependence of the
polarization angles of the binaries, based on the concise formulation given in
Cutler and Flanagan (1994). We find that the overall profiles of the PDFs are
similar for any networks composed by the second generation detectors
(Advanced-LIGO, Advanced-Virgo, KAGRA, LIGO-India). For example, 5.1% of
detected binaries would have inclination angle less than 10 degree with at most
0.1% differences between the potential networks. A perturbative expression is
also provided for generating the PDFs with a small number of parameters given
by directional averages of the quantity $\epsilon$ that characterises the
asymmetry of network sensitivities to incoming two orthogonal polarization
modes.
