Fast radio bursts ( FRB ) are millisecond-duration radio pulses with apparent extragalactic origins .
All but two of the FRBs have been discovered using the Parkes dish which employs multiple beams formed by an array of feed horns on its focal plane .
In this paper , we show that ( i ) the preponderance of multiple-beam detections , and ( ii ) the detection rates for varying dish diameters , can be used to infer the index \alpha of the cumulative fluence distribution function ( the log N -log F function : \alpha = 1.5 for a non-evolving population in a Euclidean universe ) .
If all detected FRBs arise from a single progenitor population , multiple-beam FRB detection rates from the Parkes telescope yield the constraint 0.52 < \alpha < 1.0 with 90 % confidence .
Searches at other facilities with different dish sizes refine the constraint to 0.66 < \alpha < 0.96 .
Our results favor FRB searches with smaller dishes , because for \alpha < 1 , the gain in field-of-view for a smaller dish is more important than the reduction in sensitivity .
Further , our results suggest that ( i ) FRBs are not standard candles , and ( ii ) the distribution of distances to the detected FRBs is weighted towards larger distances .
If FRBs are extragalactic , these results are consistent with a cosmological population , which would make FRBs excellent probes of the baryonic content and geometry of the Universe .