area, but since its FRB search is only restricted to the FWHM of the primary beam, the VLA non-detections cannot rule out low values of $\alpha$. The Allen Telescope Array (ATA) on the other hand is only sensitive to relatively bright events that occur close to its boresight, and the non-detections from the ATA can only rule out very flat ($\alpha\lesssim 0.6$) log$N$-log$F$ distributions.\
Since the surveys are independent trials, we multiply their respective probabilities for (non-)detections to get the aggregate probability. The posterior probability of $\alpha$ is then computed by assuming an evidence that normalizes the integral of the the aggregate probability to unity. The multi-telescope constraints on $\alpha$ thus obtained are $0.67<\alpha<1.07$ at 90% confidence, which is in excellent agreement with the Parkes multiple-beam detection constraint. Multiplying the Parkes multiple-beam and multi-telescope probability density functions, we obtained our final constraint of $0.66<\alpha<0.96$ at 90% confidence.\

![Bounds on log$N$-log$F$ parameter $\alpha$ imposed by detections and non-detections at other facilities with different aperture diameters and sensitivities: $0.67<\alpha<1.07$ (90% confidence). The probabilities have been marginalized with respect to the maximum cutoff fluence $\mathcal{F}_{\rm cut}$.\[fig:alpha\_bounds\_multitel\]](alpha_bound_multitel.pdf){width="\linewidth"}

Some caveats
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We caution the reader that the multi-telescope constraints may suffer from certain systematic errors. 