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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}

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