\label{rate}
Generally, fast radio transient detections are quantified by a local SNR...
\citet{bsb} developed a formalism for estimating the flux limit of a transient survey based on instrument properties, pointing locations, and the distribution of Galactic DM \citep{2002astro.ph..7156C}. They use the radiometer equation to estimate the flux limit for a given pointing direction. Since most transient surveys were conducted over regions with roughly similar properties (sky temperature, DM, scattering), this technique can be used to measure the mean flux limit for a given survey and estimate the apparent flux of each detection.
These estimated flux limits depend on whether the pulse has a local or extragalactic origin, since extragalactic pulses are subject to Galactic dispersion and scattering. All FRB surveys with detections and the VLA survey were made in regions of small Galactic dispersion, so the flux limits are very weakly dependent (\(<5\%\) effect) on our assumption of an extragalactic origin. Below, we assume an extragalactic origin for FRBs, consistent with the predominance of detections at high Galactic latitudes \citep{bsb,2014arXiv1405.5945P}.
With an estimate of the noise of each survey, we scale the apparent SNR to calculate the true flux. Then, for an assumed pulse width, we can estimate the range of fluxes a given survey could detect. Figure \ref{cumflux} shows the rate as a function of apparent flux. Since the primary beam introduces a fundamental ambiguity in the true FRB flux, any event could be brighter than we estimate here. On average, the whole population should be subject to the same bias, so the trend should be preserved in the limit of many FRBs.
**Fitting these distributions with a powerlaw gives...** The FRB flux distribution tends to be flatter than the Euclideans slope of –3/2. One reason may be that the burst of \citet{2007Sci...318..777L} is not a FRB, but instead a peryton \citep{bsb}. **Excluding Lorimer gives a best-fit powerlaw of...** Systematic errors make it difficult to draw a strong conclusion on whether FRBs follow a Euclidean or the Lorimer burst is an FRB. This kind of analysis will allow a more robust conclusion as the number of known FRBs grows.
The new flux limit estimates show that published rates are consistent with single population with a roughly Euclidean flux distribution. This self-consistent rate is equal to that of \citet{2013Sci...341...53T}, \(1.2\times10^{4}\) sky\(^{-1}\) day\(^{-1}\), but for a fluence limit of roughly 1.1 Jy-ms. Assuming all seven published FRB detections come from a single population, the 95% bound on the rate is 0.6–2.3\(\times10^4\) sky\(^{-1}\) day\(^{-1}\).