deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 6b9f557..6b7624a 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
...
\end{equation}  This relationship has a very small lognormal scatter of $S_{\ln\sigma | \ln M}\sim 4\%$. So:  \begin{equation}  P(\sigma | M) = \frac{1}{\sqrt{2\pi}S_{\ln\sigma | \ln M}} exp\left({\frac{(\ln\sigma \exp\left({\frac{(\ln\sigma  - \ln\langle \sigma | M \rangle)^{2}}{2 S_{\ln\sigma | \ln M}^{2}}}\right ) \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: 
...
\end{equation}  Really there are completeness and purity terms in there as well, but lets ignore those for a second. So that is our expected distribution of $\hat{\sigma}$ for a given $M$. The other term is equally important $P(\hat{\sigma} | \sigma)$. This represents the probability of observing a velocity dispersion $\hat{\sigma}$ given $\sigma$. Why is this important? When we observe clusters in the real universe, we don't measure the ``Evrard" velocity dispersion $\sigma$. We are randomly drawing from a distribution where the $\sigma$ is the expectation value. This is what \citet{Gifford13a} means by l.o.s effects. So what is that distribution? It's approximately lognormal with $S_{\log(\hat{\sigma}) | \log(\sigma)}\sim 25\%$. So:  \begin{equation}  P(\hat{\sigma} | \sigma) = \frac{1}{\sqrt{2\pi}S_{\ln\hat{\sigma} | \ln\sigma}} e^{\frac{(\ln\hat{\sigma} \exp\left({\frac{(\ln\hat{\sigma}  - \ln\sigma)^{2}}{2 S_{\ln\hat{\sigma} | \ln\sigma}^2}} \ln\sigma}^2}}\right)  \end{equation}  But we are binning! That means that we have a distribution of masses in our bin that we must integrate over. This integral takes the form: