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We analytically discuss probability distribution function (PDF) forinclinations of merging compact binaries whose gravitational waves arecoherently detected by a network of ground based interferometers. The PDF wouldbe useful for studying prospects of (1) simultaneously detectingelectromagnetic signals (such as gamma-ray-bursts) associated with binarymergers and (2) statistically constraining the related theoretical models fromthe actual observational data of multi-messenger astronomy. Our approach issimilar to Schutz (2011), but we explicitly include the dependence of thepolarization angles of the binaries, based on the concise formulation given inCutler and Flanagan (1994). We find that the overall profiles of the PDFs aresimilar for any networks composed by the second generation detectors(Advanced-LIGO, Advanced-Virgo, KAGRA, LIGO-India). For example, 5.1% ofdetected binaries would have inclination angle less than 10 degree with at most0.1% differences between the potential networks. A perturbative expression isalso provided for generating the PDFs with a small number of parameters givenby directional averages of the quantity $\epsilon$ that characterises theasymmetry of network sensitivities to incoming two orthogonal polarizationmodes.

Probability distribution function for inclinations of merging compact binaries detected by gravitational wave interferometers