at most, about 500 Jy ms (Ravi et al., in prep.). We thus marginalize all probabilities derived in this section over the cut-off fluence while assuming a uniform prior between 0.5 and 50 kJy ms.\
Fig. \[fig:alpha\_bounds\_multitel\] shows the probability density function of $\alpha$ evaluated using Equation \[eqn:ndet\] for the various surveys whose parameters are given in Table 2. In doing so we have assumed Poisson statistics for the arrival of FRBs, i.e if the expected numbers of events for a survey is $N_{\rm det}$, then the probability of discovering $N_{\rm event}$ events in a survey is $$\mathcal{P}(N_{\rm event},N_{\rm det})  = \frac{N_{\rm det}^{N_{\rm event}} {\rm e}^{-N_{\rm det}}}{N_{\rm event}!}.$$ As seen in the Figure, the strongest constraints on $\alpha$ come from the Arecibo telescope, owing to its excellent sensitivity afforded by the large collecting area. For very flat log$N$-log$F$ distributions ($\alpha\lesssim 0.5$), we expect a large number of bright events which will be detected even in the sidelobes of the Arecibo’s beam pattern. This partially offsets the small FOV of the Arecibo dish. Steeper log$N$-log$F$ distributions ($\alpha\gtrsim 1.1$) simply yield a large number of faint events which will cross Arecibo’s detection threshold around boresight. The VLA also has a large collecting 