consider the sensitivity of our results to variations in the source-count normalization in Section 4.2. Table 2 summarizes the various telescope parameters for the different published results we consider here; for details, see Appendix A2. To design a common algorithm to compute the necessary statistics, we have absorbed telescope efficiency parameters into the system temperature, $T_{\rm sys}$, so that the final thermal noise per time-integration in our formalism matches the values quoted.\

Detection rates
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We assume that FRBs have a mean duration of $\tau_{\rm FRB}=3$ ms, a mean DM of 780 pc cm$^{-3}$ [@law2015], and no correlation between the two quantities. If $\Delta\nu$ is the bandwidth, $d$ is the aperture diameter, $\eta$ is the aperture efficiency, and $\tau_{\rm int}$ is the spectrometer integration time, then the thermal noise flux-density in a single time-integration for an incoherent summation of signals from $N_{\rm ant}$ antennas is $$\label{eqn:th_noise}
S_{\rm th} = \frac{8k_{\rm B}T_{\rm sys}}{\eta\pi d^2}\frac{1}{\sqrt{2N_{\rm ant}\Delta\nu\tau_{\rm int}}},$$ where $k_{\rm B}$ is Boltzmann’s constant. The threshold for detection depends on the amount of dispersion smearing, the temporal width of the burst with respect to the integration time, and the threshold used for detection (number of $\sigma$ above thermal noise) $\zeta$. The fluence threshold 