(CLF) formalism introduced by @yang (see also @vdb and references therein) and by Cooray and collaborators (e.g., @cooray [@coorayc] and references therein). This consists of replacing a deterministic mass-luminosity relation, like the one in equation (\[mlfit\]), with a distribution of luminosity around an average value for any given halo mass, $\phi_{CLF}(L|M)dL$, which represents the probability of having a galaxy of luminosity $L$ in a halo of mass $M$. Note that here we are only applying this to the central galaxy in any given halo, since that is the important one for the study of BCGs; the distribution of satellite galaxies we draw directly from the distribution of subhalo masses.

An important point is that this CLF must, by definition, match the observed luminosity function when it is integrated over all haloes, i.e. $\int_0^\infty \phi_{CLF}(L|M) n(M) dM=\phi(L)$, where $n(M)$ is the halo mass function, $\phi(L)$ the observed luminosity function and $L$ should only be considered in the range where the haloes dominate the number of hosts (i.e., at high luminosity, which is precisely the range we are interested in when looking at BCGs;otherwise, we would also need to account for subhalo contribution). The introduction of scatter then leads to a problem 