priori*]{} reason to assume any specific value for the dispersion. Furthermore, this value is linked to the modified mass-luminosity relation of equation (\[eq:mlmod\]), so it needs to be defined in some way in order to determine the latter. We do this by determining which value of the dispersion leads to an average luminosity as a function of mass which best fits observational values. For simplicity, we will consider that the value of the dispersion, $\sigma$, is constant and independent of mass. Since we are only interested in the bright end, where we expect central galaxies to dominate, and these are known to have only a small scatter in luminosity (e.g., @pl [@bernardi]), this is quite likely a good approximation. Semi-analytical modelling also shows that scatter in stellar mass is only a weak function of halo mass [@wang].

The luminosity $L$ of a galaxy in a host of mass $M$, is then given by a lognormal distribution of the type:

$$\label{eq:lognormal}
\phi_{CLF}(L|M)dL=\frac{1}{\sqrt{2\pi}\sigma_{LN} L}{\rm exp}
\Big(-\frac{({\rm ln} [L/L_0(M)]+\sigma_{LN}^2/2)^2}{2 \sigma_{LN}^2} \Big) dL \, ,$$

where $L_0(M)$ is some average luminosity for a host of mass $M$ (discussed further below), and $\sigma$ is the dispersion in the normal logarithm of the luminosity. Note that 