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Suppose there are astrophysical events that occur at random places in space. We denote the rate of such events per unit volume by $ \dot{n} $. Suppose we observe for a time $ T $ such that the density of events during that time $ n = \dot{n} T $. Suppose further that the luminosity of each event is the same $ L $. The observed flux from each event would be

$ f = \frac{L}{4 \pi r^2} $

where $ r $ is the distance to the observer. The number distribution of the fluxes would be

$ dN = 4 \pi r^2 dr \cdot n \propto f^{-5/2} df $

If the threshold for detection is $ f_{th} $, then the probability for each flux is

$ \frac{dP}{dx} = -\frac{2}{3} x^{-5/2} $

where $ x = \frac{f}{f_{th}} $ is the normalised flux.

For each event there's a maximal distance from which it can be detected

$ r_{th} = \sqrt{\frac{L}{4 \pi f}} $

From each radius it is possible to construct a sphere around the observer. The ratio between the volume of the sphere from which an event was detected, and that of a sphere of maximum distance is $ x^{-3/2} $. On average, it should be $ \frac{1}{2} $ because an event has an equal opportunity to be in either half of the sphere. This can also be shown mathematically

$ < x^{-3/2} > = \int_1^{\infty} x^{-3/2} dP = \frac{3}{2} \int_1^{\infty} x^{-4} dx = \frac{3}{2} \left[\frac{1}{3} x^{-3}\right]_1^{\infty} = \frac{1}{2} $