deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index ceab047..75f0c0e 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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There are several key probabilities we need to know in order to accurately predict the expectation value of some observable with mass. In more statistical language, we would like to 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. \citet{Evrard08} \cite{Evrard08}  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}  \langle \sigma | M \rangle = 1093 \left(\frac{h(z) M}{1e15 M_{\odot}}\right)^{0.34}  \end{equation}