Suppose there are astrophysical events that occur at random places in space. We denote the rate of such events per unit volume by . Suppose we observe for a time such that the density of events during that time . Suppose further that the luminosity of each event is the same . The observed flux from each event would be
where is the distance to the observer. The number distribution of the fluxes would be
If the threshold for detection is , then the probability for each flux is
where is the normalised flux.
For each event there's a maximal distance from which it can be detected
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 . On average, it should be because an event has an equal opportunity to be in either half of the sphere. This can also be shown mathematically