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There are several key probabilities we need to know in order to accurately predict the scatter/bias of some observable with mass. In more statistical language, we must know $P(\hat{\theta}|M)$. That is, given a cluster of mass $M$, what is the probability the observable is detected as $\hat{\theta}$. For example, the observable $\hat{\theta}$ can be velocity dispersion, some richness, or total luminosity.   Let's take velocity dispersion as an example here. There are two relevant velocity dispersions in our studies. The first is the true 3D/2D velocity dispersion $\sigma$ which we cannot measure in the real universe. \cite{doi:10.1086/521616} \cite{2008ApJ...672..122E}  measured $\sigma$ for halos in N-body simulations and showed that they relate to the critical mass $M_{200}$ of the host halo on a very tight relation \begin{equation}  \end{equation}