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\end{equation}  Really there are completeness and purity terms in there as well, but lets ignore those for a second. So that is our expected distribution for a given $M$, but we are binning! That means that we have a distribution of masses in our bin that we must integrate over. What does this integral look like?  \begin{equation}  \langle \hat{\sigma} \rangle = \int_{min(bin)}^{max_bin} \int_{min(bin)}^{max(bin)}  dM \frac{d \langle n \rangle}{dM} P(\hat{\sigma} | M) \end{equation}