tokens: event (90% and 99% Poisson errors shown in blue). For this figure we assumed that the population in total produces 10 neutrino events (within a given time). While the selection bias is strong for sources detected with a single event, it quickly vanishes as soon as three or more events are detected from the same source. The effective densities of several source classes are indicated (compare Table 1 ). For each source class two different luminosity functions were adopted. The numbers in the upper line are for equally luminous sources, while a lognormal distribution with a width of one order of magnitude was assumed for the second line to show the impact of large luminosity fluctuations between individual sources. This variation in luminosity has the same impact on our results as using the observed distribution of gamma-ray luminosities for Fermi /LAT blazars which stretches over five orders of magnitudes (see Fig. 2 in Ajello et al. 2014 ) and also reproduces fluctuations observed between individual gamma-ray burst by the Swift Burst Alert Telescope ( Wanderman Piran 2010 ). This simplistic treatment does not account for correlations between the source redshift and luminosity, which have been observed for example in galaxy clusters ( Gruppioni et al. 2013 ). We find that the effect of the different redshift distributions and luminosity functions can be taken into account by using an effective density. The effective density is the density of a population of equally luminous sources without cosmic evolution which reproduces an equally large bias, i.e., the same median value. A similar calculation is performed in Murase Waxman (2016) , where the effective density quantifies the effect of the luminosity function, but not of the source evolution. The last column of Table 1 shows that the rather large luminosity fluctuations between individual sources reduce the effective source density by about one order of magnitude since a large fraction of the total neutrino flux is produced by a smaller number of sources. A detector that is not equally sensitive to all parts of the sky can be considered in the same way as the luminosity fluctuations since sources in the favored direction appear to be brighter. For example, if the detector is only sensitive in half of the sky, the effective density is reduced by a factor of two and the bias is correspondingly smaller. Figure 2 shows the size of the bias for different source densities assuming equally luminous sources and no cosmic evolution. The red bands show the expected neutrino flux of the sources where the dotted line indicates the median flux and the shaded area contains 90% and 99% of the probability distribution. The effective densities of the populations listed in Table 1 are indicated in Fig. 2 ; the width of the inner red band approximately reproduces the size of the 90% regions that were obtained for the different redshift distributions (see Table 1 ). Especially for sources detected with just a
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