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We present analytical solutions for the integral distribution of arbitrarybursting or steady source counts as a function of peak photon count rate withinFriedmann cosmological models. We discuss both the standard candle andtruncated power-law luminosity function cases with a power-law densityevolution. While the analysis is quite general, the specific example discussedhere is that of a cosmological gamma-ray burst distribution. These solutionsshow quantitatively the degree of dependence of the counts on the density andluminosity function parameters, as well as the the weak dependence on theclosure parameter and the maximum redshift. An approximate comparison with thepublicly available Compton Gamma Ray Observatory data gives an estimate of themaximum source luminosity and an upper limit to the minimum luminosity. Wediscuss possible ways of further constraining the various parameters.Comment: 16 pages plus one figure, uuencoded postscript file. to appear in  Ap.J

The Brightness Distribution of Bursting Sources in Relativistic Cosmologies