deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 0f776d1..952b2a2 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
\begin{equation}  \langle \sigma | M \rangle = 1093 \left(\frac{h(z) M}{1e15 M_{\odot}}\right)^{0.34}  \end{equation}  This relationship has a very small lognormal scatter of $S_{\ln\sigma | \lnM}\sim \ln M}\sim  4\%$. So: \begin{equation}  P(\sigma | M) = \frac{1}{\sqrt{2\pi}S_{\ln\sigma | \lnM}} \ln M}}  e^{\frac{(\ln\sigma - \ln\langle \sigma | M \rangle)^{2}}{2 S_{\ln\sigma | \lnM}^{2}}} \ln M}^{2}}}  \end{equation}  The second velocity dispersion is the observed velocity dispersion $\hat{\sigma}$. In \citet{Gifford13a}, we define this as the l.o.s velocity dispersion. This has all kinds of ugly things in it including cluster shape effects, cluster environment contamination, substructure, redshift-space interlopers, and non-gaussianity. Not to mention the low number statistics that exist at low mass. Even though this is a messy observable, most are, and this is what we need to predict for a given mass $M$. So here is the generative model for observable: