mean DM to be 375 pc cm$^{-3}$ or 1600 pc cm$^{-3}$. The respective constraints on $\alpha$ are $0.68<\alpha<1.07$ and $0.66<\alpha<1.06$.

Hence, our constraints are robust to even $100$% changes in the assumed FRB-rate normalization, mean FRB width and DM. We do however note that a drastic reduction in the all sky FRB rate by a factor of $\sim 10$ yields values of $\alpha$ that are roughly consistent with a Euclidean population.\
In addition to burst properties, there may be systematic effects due to practical choices in experimental design and detection algorithms. The Arecibo search for FRBs for instance, was limited to low Galactic latitudes where observed FRB fluences may be significantly lower due to scintillation-induced biases [@macquart2015].[^4] In addition, since the Arecibo FRB search was limited to DM less than 2000 pc cm$^{-3}$ [@spitler2014], weaker events that preferentially originate from larger distances may have been overlooked. Finally, as single-burst detection techniques are evolving, different surveys (even on the same telescope) may be employing algorithms with different missed-detection and false-positive rates which makes it difficult to bring their (non-)detections into a common probabilistic framework. Nevertheless, the robustness of our constraints on $\alpha$ against large variations in the event-rate normalization and mean 