care must be taken when looking at the calculated average mass-luminosity relation. In our approach, the average luminosity at fixed mass needs to go down, relative to the scatter-less case or, looking at it the opposite way, the same average luminosity is obtained for higher mass haloes. This is due to the fact that, when considering the CLF, we are doing the binning by mass (or more precisely, taking the conditional variable in the distribution to be the mass). If we had instead binned by luminosity, the effect of introducing scatter would have been the opposite: the average mass correspoding to a given luminosity would instead have gone down. This is to be expected and is just a statistical effect of the two different ways in which the conditional function can be defined. It does however mean that care must be taken when comparing results of different authors to look at how the binning was done in each case.

In this paper we will consider two different models for the CLF of the central galaxy: a simpler model where we assume the distribution is lognormal, but where we are left with a free parameter in the scatter introduced; and a 