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