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.
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