Consider a homogeneous distribution of GRBs of luminosity L
with a space density of n
For a spherical shell of these sources a distance r from us:
Since this proportionality is independent from the luminosity L,
it will also hold for any distribution of intrinsic luminosities for the GRBs
(as long as the distribution does not change with distance from us).
A plot of logN versus logf should hence produce a straight line
with a slope of -1.5 if the sources of the GRBs are distributed uniformly through space.
In fact, BATSE observations of GRBs reveal a departure from such a relation: