Statistical properties of the combined emission of a population of discrete sources: astrophysical implications

We study the statistical properties of the combined emission of a population of discrete sources (for example, the X-ray emission of a galaxy due to its population of X-ray binaries). Namely, we consider the dependence of their total luminosity L-tot = SigmaL(k) and of the fractional rms(tot) of their variability on the number of sources n or, equivalently, on the normalization of the luminosity function. We show that, as a result of small number statistics, a regime exists in which L-tot grows non-linearly with n, in apparent contradiction with the seemingly obvious prediction L-tot = L (dN/dL) dL proportional to n. In this non-linear regime, rms(tot) decreases with n significantly more slowly than expected from the rms proportional to 1/ rootn averaging law. For example, for a power-law luminosity function with a slope of alpha = 3 2, in the non-linear regime, L-tot proportional to n(2) and rms(tot) does not depend at all on the number of sources n. Only in the limit of n --> infinity do these quantities behave as intuitively expected, L-tot proportional to n and rms(tot) proportional to1/rootn. We give exact solutions and derive convenient analytical approximations for L-tot and rms(tot). Using the total X-ray luminosity of a galaxy due to its X-ray binary population as an example, we show that the L-X-star formation rate and L-X-M-* relations predicted from the respective 'universal' luminosity functions of high- and low-mass X-ray binaries are in good agreement with observations. Although caused by small number statistics, the non-linear regime in these examples extends as far as SFR less than or similar to 4-5 M. yr(-1) and log(M-*/M.) less than or similar to 10.0-10.5, respectively.