[spec] additions to complex graphs and exp identities

diff --git a/methods/statistics-ref.tex b/methods/statistics-ref.tex
index 87319996ce78586398711d71132ff817f9babf6d..ae4153ec75e8a4c1656ac484a741d36371e9b45f 100644 (file)
--- a/methods/statistics-ref.tex
+++ b/methods/statistics-ref.tex
@@ -24,11 +24,17 @@ Independent events:
 
 \subsection*{Combinatorics}
 
-\begin{itemize} \tightlist
-  \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
-  \item \colorbox{highlight}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
-  \item Note \({n \choose k} = {n \choose k-1}\)
-\end{itemize}
+\begin{align*}
+  \text{Arrangements} && {n \choose k} & = \frac{n!}{(n-k)} \\
+  \text{Combinations} && {n \choose k} & = \frac{n!}{k!(n-k)!}
+\end{align*}
+
+Note \({n \choose k} = {n \choose k-1}\)
+
+\begin{cas}
+  Keyboard \(\rightarrow\) Advance \(\rightarrow\) \keystroke{nCr}/\keystroke{nPr} \\
+  \-\hspace{1em} \texttt{nCr(n, r)} or \texttt{nPr(n, r)}
+\end{cas}
 
 \subsection*{Distributions}
 
@@ -151,22 +157,6 @@ A continuous random variable \(X\) has a pdf \(f\) such that:
 
 \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
 
-\begin{cas}
-  Define piecewise functions: \\
-  \-\hspace{1em}Math3 \(\rightarrow\)
-  \begin{tikzpicture}%
-    \draw rectangle (0.5,0.5); 
-    \node at (0.08,0.25) {\(\{\)};
-    \filldraw [black] (0.15, 0.4) rectangle(0.25, 0.3);
-    \draw (0.35, 0.4) rectangle(0.45, 0.3);
-    \node [font=\footnotesize] at (0.3,0.3) {\verb;,;};
-    \draw (0.15, 0.2) rectangle(0.25, 0.1);
-    \node [font=\footnotesize] at (0.3,0.1) {\verb;,;};
-    \draw (0.35, 0.2) rectangle(0.45, 0.1);
-  \end{tikzpicture}
-  % TODO: finish this section
-\end{cas}
-
 \subsection*{Two random variables \(X, Y\)}
 
 If \(X\) and \(Y\) are independent: