From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 9 Sep 2019 11:58:37 +0000 (+1000)
Subject: [methods] clean up
X-Git-Tag: yr12~35
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/25e76956cff9c6a5aee17fba3a00c1e8fd4fc330

[methods] clean up
---

diff --git a/methods/calculus.pdf b/methods/calculus.pdf
new file mode 100644
index 0000000..d48665f
Binary files /dev/null and b/methods/calculus.pdf differ
diff --git a/methods/calculus.tex b/methods/calculus.tex
index 6f57dc0..1896189 100644
--- a/methods/calculus.tex
+++ b/methods/calculus.tex
@@ -1,3 +1,7 @@
+\documentclass[methods-collated.tex]{subfiles}
+
+\begin{document}
+
 \section{Calculus}
 
 \subsection*{Average rate of change}
@@ -19,16 +23,12 @@
 \subsection*{Limit theorems}
 
 \begin{enumerate}
-\def\labelenumi{\arabic{enumi}.}
-\tightlist
-\item
-  For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
-\item
-  \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
-\item
-  \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
-\item
-  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
+    \def\labelenumi{\arabic{enumi}.}
+    \tightlist
+  \item For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
+  \item \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
+  \item \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
+  \item \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 \end{enumerate}
 
 A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
@@ -39,13 +39,10 @@ A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
 
 Not differentiable at:
 \begin{itemize}
-\tightlist
-\item
-  discontinuous points
-\item
-  sharp point/cusp
-\item
-  vertical tangents (\(\infty\) gradient)
+    \tightlist
+  \item discontinuous points
+  \item sharp point/cusp
+  \item vertical tangents (\(\infty\) gradient)
 \end{itemize}
 
 \subsection*{Tangents \& gradients}
@@ -64,25 +61,20 @@ Not differentiable at:
 For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
 
 \begin{itemize}
-\tightlist
-\item
-  \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
-\item
-  \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
-\item
-  Endpoints are included, even where gradient \(=0\)
+    \tightlist
+  \item \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
+  \item \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
+  \item Endpoints are included, even where gradient \(=0\)
 \end{itemize}
 
-\columnbreak
-
-\subsubsection*{Solving on CAS}
-
-\colorbox{cas}{\textbf{In main}}: type function. Interactive
-\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal
-\textbar{} Tan line)\\
-\colorbox{cas}{\textbf{In graph}}: define function. Analysis
-\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line).
-Type \(x\) value to solve for a point. Return to show equation for line.
+\begin{cas}
+  \colorbox{cas}{\textbf{In main}}: type function. Interactive
+  \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal
+  \textbar{} Tan line)\\
+  \colorbox{cas}{\textbf{In graph}}: define function. Analysis
+  \(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line).
+  Type \(x\) value to solve for a point. Return to show equation for line.
+\end{cas}
 
 \subsection*{Stationary points}
 
@@ -91,81 +83,81 @@ Type \(x\) value to solve for a point. Return to show equation for line.
   \textbf{Point of inflection:} && f^{\prime\prime} &= 0
 \end{align*}
 
-                  \begin{tikzpicture}
-                    \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
-                      \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)};
-                      \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)};
-                      \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
-                      \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
-                      \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
-                    \end{axis}
-                  \end{tikzpicture}\\
-                  \begin{tikzpicture}
-                    \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
-                      \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)};
-                      \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)};
-                      \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
-                    \end{axis}
-                  \end{tikzpicture}\\
-\pagebreak
+\begin{tikzpicture}
+  \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
+    \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)};
+    \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)};
+    \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
+    \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
+    \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
+  \end{axis}
+\end{tikzpicture}\\
+\begin{tikzpicture}
+  \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
+    \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)};
+    \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)};
+    \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
+  \end{axis}
+\end{tikzpicture}
+
 \subsection*{Derivatives}
 
-\definecolor{shade1}{HTML}{ffffff}
-\definecolor{shade2}{HTML}{F0F9E4}
-\rowcolors{1}{shade1}{shade2}
-                  \renewcommand{\arraystretch}{1.4}
-                  \begin{tabularx}{\columnwidth}{rX}
-                    \hline
-                    \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
-                    \hline
-                    \(\sin x\) & \(\cos x\)\\
-                    \(\sin ax\) & \(a\cos ax\)\\
-                    \(\cos x\) & \(-\sin x\)\\
-                    \(\cos ax\) & \(-a \sin ax\)\\
-                    \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
-                    \(e^x\) & \(e^x\)\\
-                    \(e^{ax}\) & \(ae^{ax}\)\\
-                    \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
-                    \(\log_e x\) & \(\dfrac{1}{x}\)\\
-                    \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
-                    \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
-                    \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
-                    \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
-                    \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
-                    \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
-                    \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
-                    \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
-                    \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
-                    \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
-                    \hline
-                  \end{tabularx}
-                  \vspace*{\fill}
-                  \columnbreak
+\rowcolors{1}{white}{orange}
+\renewcommand{\arraystretch}{1.4}
+
+\begin{tabularx}{\columnwidth}{rX}
+  \hline
+  \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
+  \hline
+  \(\sin x\) & \(\cos x\)\\
+  \(\sin ax\) & \(a\cos ax\)\\
+  \(\cos x\) & \(-\sin x\)\\
+  \(\cos ax\) & \(-a \sin ax\)\\
+  \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
+  \(e^x\) & \(e^x\)\\
+  \(e^{ax}\) & \(ae^{ax}\)\\
+  \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
+  \(\log_e x\) & \(\dfrac{1}{x}\)\\
+  \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
+  \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
+  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
+  \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
+  \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
+  \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
+  \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
+  \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
+  \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
+  \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
+  \hline
+\end{tabularx}
+
 \subsection*{Antiderivatives}
-\rowcolors{1}{shade1}{cas}
-                  \renewcommand{\arraystretch}{1.4}
-                  \begin{tabularx}{\columnwidth}{rX}
-                    \hline
-                    \(f(x)\) & \(\int f(x) \cdot dx\) \\
-                    \hline
-                    \(k\) (constant) & \(kx + c\)\\
-                    \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
-                    \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
-                    \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
-                    \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
-                    \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
-                    \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
-                    \(e^k\) & \(e^kx + c\)\\
-                    \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
-                    \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
-                    \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
-                    \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
-                    \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
-                    \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
-                    \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
-                    \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
-                    \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
-                    \hline
-                  \end{tabularx}
-\vspace*{\fill}
-                  \columnbreak
+
+\rowcolors{1}{white}{lblue}
+\renewcommand{\arraystretch}{1.4}
+
+\begin{tabularx}{\columnwidth}{rX}
+  \hline
+  \(f(x)\) & \(\int f(x) \cdot dx\) \\
+  \hline
+  \(k\) (constant) & \(kx + c\)\\
+  \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
+  \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
+  \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
+  \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
+  \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
+  \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
+  \(e^k\) & \(e^kx + c\)\\
+  \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
+  \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
+  \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
+  \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
+  \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
+  \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
+  \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
+  \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
+  \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
+  \hline
+\end{tabularx}
+
+\end{document}
diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf
index 12df5ce..6c67429 100644
Binary files a/methods/methods-collated.pdf and b/methods/methods-collated.pdf differ
diff --git a/methods/methods-collated.tex b/methods/methods-collated.tex
index b064130..42fac1d 100644
--- a/methods/methods-collated.tex
+++ b/methods/methods-collated.tex
@@ -4,15 +4,16 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{blindtext}
+\usepackage{dblfloatfix}
 \usepackage{enumitem}
 \usepackage{fancyhdr}
 \usepackage[a4paper,margin=2cm]{geometry}
 \usepackage{graphicx}
 \usepackage{harpoon}
 \usepackage{listings}
-\usepackage{longtable}
 \usepackage{makecell}
 \usepackage{mathtools}
+\usepackage{mathtools}
 \usepackage{multicol}
 \usepackage{multirow}
 \usepackage{newclude}
@@ -32,17 +33,30 @@
 
 \usetikzlibrary{%
   angles,
+  arrows,
+  arrows.meta,
   calc,
   datavisualization.formats.functions,
   decorations,
   decorations.markings,
+  decorations.text,
   decorations.pathreplacing,
   decorations.text,
   scopes
 }
+
 \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+
 \usepgflibrary{arrows.meta}
-\pgfplotsset{compat=1.6}
+\pgfplotsset{compat=1.16}
+\pgfplotsset{every axis/.append style={
+  axis x line=middle,    % centre axes
+  axis y line=middle,
+  axis line style={->},  % arrows on axes
+  xlabel={$x$},          % axes labels
+  ylabel={$y$}
+}}
+
 \psset{dimen=monkey,fillstyle=solid,opacity=.5}
 \def\object{%
     \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
@@ -51,29 +65,23 @@
         \psline{->}(0,-2)%
         \uput[-90]{*0}(0,-2){$\vec{w}$}}
 }
-\newcommand{\tg}{\mathop{\mathrm{tg}}}
-\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
-\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
-\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
-\pgfplotsset{every axis/.append style={
-  axis x line=middle,    % centre axes
-  axis y line=middle,
-  axis line style={->},  % arrows on axes
-  xlabel={$x$},          % axes labels
-  ylabel={$y$}
-}}
 
 \pagestyle{fancy}
 \fancyhead[LO,LE]{Year 12 Methods}
 \fancyhead[CO,CE]{Andrew Lorimer}
 \fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
 
+\newcommand{\tg}{\mathop{\mathrm{tg}}}
+\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
+\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
+\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
+
 \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
+\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
 \linespread{1.5}
 \setlength{\parindent}{0cm}
 \setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
-\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
-\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
 
 \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}
 \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}
@@ -82,9 +90,8 @@
 \definecolor{cas}{HTML}{e6f0fe}
 \definecolor{important}{HTML}{fc9871}
 \definecolor{dark-gray}{gray}{0.2}
-\definecolor{shade1}{HTML}{ffffff}
-\definecolor{shade2}{HTML}{e6f2ff}
-\definecolor{shade3}{HTML}{cce2ff}
+\definecolor{orange}{HTML}{e6beb2}
+\definecolor{lblue}{HTML}{e5e9f0}
 
 \newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
 \newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
diff --git a/methods/statistics-ref.pdf b/methods/statistics-ref.pdf
new file mode 100644
index 0000000..2cbc960
Binary files /dev/null and b/methods/statistics-ref.pdf differ
diff --git a/methods/statistics-ref.tex b/methods/statistics-ref.tex
index 3461bce..4a08282 100644
--- a/methods/statistics-ref.tex
+++ b/methods/statistics-ref.tex
@@ -1,335 +1,261 @@
 \documentclass[methods-collated.tex]{subfiles}
+
 \begin{document}
-  \section{Statistics}
 
-  \subsection*{Probability}
+\section{Statistics}
 
-  \begin{align*}
-    \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
-    \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
-    \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
-    \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
-  \end{align*}
+\subsection*{Probability}
 
-  Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
+\begin{align*}
+  \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
+  \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
+  \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
+  \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
+\end{align*}
 
-  Independent events:
-  \begin{flalign*}
-    \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
-    \Pr(A|B) &= \Pr(A) \\
-    \Pr(B|A) &= \Pr(B)
-  \end{flalign*}
+Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
 
-  \subsection*{Combinatorics}
+Independent events:
+\begin{flalign*}
+  \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
+  \Pr(A|B) &= \Pr(A) \\
+  \Pr(B|A) &= \Pr(B)
+\end{flalign*}
 
-  \begin{itemize}
-    \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
-    \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
-    \item Note \({n \choose k} = {n \choose k-1}\)
-  \end{itemize}
+\subsection*{Combinatorics}
 
-  \subsection*{Distributions}
+\begin{itemize}
+  \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
+  \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
+  \item Note \({n \choose k} = {n \choose k-1}\)
+\end{itemize}
 
-  \subsubsection*{Mean \(\mu\)}
+\subsection*{Distributions}
 
-  \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
+\subsubsection*{Mean \(\mu\)}
 
-  \begin{align*}
-    E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
-    &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
-    &= \int_\textbf{X} (x \cdot f(x)) \> dx
-  \end{align*}
+\textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
 
-  \subsubsection*{Mode}
+\begin{align*}
+  E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
+  &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
+  &= \int_\textbf{X} (x \cdot f(x)) \> dx
+\end{align*}
 
-  Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
+\subsubsection*{Mode}
 
-  \subsubsection*{Median}
+Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.
 
-  If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.
+\subsubsection*{Median}
 
-  \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
+If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.
 
-  \subsubsection*{Variance \(\sigma^2\)}
+\[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
 
-  \begin{align*}
-    \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
-    &= \sum (x-\mu)^2 \times \Pr(X=x) \\
-    &= \sum x^2 \times p(x) - \mu^2 \\
-    &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
-    &= E\left[(X-\mu)^2\right]
-  \end{align*}
+\subsubsection*{Variance \(\sigma^2\)}
 
-  \subsubsection*{Standard deviation \(\sigma\)}
+\begin{align*}
+  \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
+  &= \sum (x-\mu)^2 \times \Pr(X=x) \\
+  &= \sum x^2 \times p(x) - \mu^2 \\
+  &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2 \\
+  &= E\left[(X-\mu)^2\right]
+\end{align*}
 
-  \begin{align*}
-    \sigma &= \operatorname{sd}(X) \\
-    &= \sqrt{\operatorname{Var}(X)}
-  \end{align*}
+\subsubsection*{Standard deviation \(\sigma\)}
 
-  \subsection*{Binomial distributions}
+\begin{align*}
+  \sigma &= \operatorname{sd}(X) \\
+  &= \sqrt{\operatorname{Var}(X)}
+\end{align*}
 
-  Conditions for a \textit{binomial distribution}:
-  \begin{enumerate}
-    \item Two possible outcomes: \textbf{success} or \textbf{failure}
-    \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
-    \item Finite number \(n\) of independent trials
-  \end{enumerate}
+\subsection*{Binomial distributions}
 
+Conditions for a \textit{binomial distribution}:
+\begin{enumerate}
+  \item Two possible outcomes: \textbf{success} or \textbf{failure}
+  \item \(\Pr(\text{success})\) (=\(p\)) is constant across trials
+  \item Finite number \(n\) of independent trials
+\end{enumerate}
 
-  \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
 
-  \begin{align*}
-    \mu(X) &= np \\
-    \operatorname{Var}(X) &= np(1-p) \\
-    \sigma(X) &= \sqrt{np(1-p)} \\
-    \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
-  \end{align*}
+\subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
 
-  \begin{cas}
-    Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input
-    \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
-      \item [x:] no. of successes
-      \item [numtrial:] no. of trials
-      \item [pos:] probability of success
-    \end{description}
-  \end{cas}
+\begin{align*}
+  \mu(X) &= np \\
+  \operatorname{Var}(X) &= np(1-p) \\
+  \sigma(X) &= \sqrt{np(1-p)} \\
+  \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
+\end{align*}
 
-  \subsection*{Continuous random variables}
+\begin{cas}
+  Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf;
+  \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
+    \item [x:] no. of successes
+    \item [numtrial:] no. of trials
+    \item [pos:] probability of success
+  \end{description}
+\end{cas}
 
-  A continuous random variable \(X\) has a pdf \(f\) such that:
+\subsection*{Continuous random variables}
 
-  \begin{enumerate}
-    \item \(f(x) \ge 0 \forall x \)
-    \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
-  \end{enumerate}
+A continuous random variable \(X\) has a pdf \(f\) such that:
 
-  \begin{align*}
-    E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
-    \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
-  \end{align*}
+\begin{enumerate}
+  \item \(f(x) \ge 0 \forall x \)
+  \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
+\end{enumerate}
 
-  \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
+\begin{align*}
+  E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
+  \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
+\end{align*}
 
+\[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
 
-  \subsection*{Two random variables \(X, Y\)}
 
-  If \(X\) and \(Y\) are independent:
-  \begin{align*}
-    \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
-    \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
-  \end{align*}
+\subsection*{Two random variables \(X, Y\)}
 
-  \subsection*{Linear functions \(X \rightarrow aX+b\)}
+If \(X\) and \(Y\) are independent:
+\begin{align*}
+  \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
+  \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
+\end{align*}
 
-  \begin{align*}
-    \Pr(Y \le y) &= \Pr(aX+b \le y) \\
-    &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
-    &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
-  \end{align*}
+\subsection*{Linear functions \(X \rightarrow aX+b\)}
 
-  \begin{align*}
-    \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
-    \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
-  \end{align*}
+\begin{align*}
+  \Pr(Y \le y) &= \Pr(aX+b \le y) \\
+  &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
+  &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
+\end{align*}
 
-  \subsection*{Expectation theorems}
+\begin{align*}
+  \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
+  \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
+\end{align*}
 
-  For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
+\subsection*{Expectation theorems}
 
-  \begin{align*}
-    E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
-    E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
-    &\ne [E(X)]^n \\
-    E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
-    E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
-    E(X+Y) &= E(X) + E(Y) \tag{two variables}
-  \end{align*}
+For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
 
-  \subsection*{Sample mean}
+\begin{align*}
+  E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
+  E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
+  &\ne [E(X)]^n \\
+  E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
+  E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
+  E(X+Y) &= E(X) + E(Y) \tag{two variables}
+\end{align*}
 
-  Approximation of the \textbf{population mean} determined experimentally.
+\begin{figure*}[hb]
+  \centering
+  \include{../spec/normal-dist-graph}
+\end{figure*}
 
-  \[ \overline{x} = \dfrac{\Sigma x}{n} \]
+\subsection*{Sample mean}
 
-  where
-  \begin{description}[nosep, labelindent=0.5cm]
-    \item \(n\) is the size of the sample (number of sample points)
-    \item \(x\) is the value of a sample point
-  \end{description}
+Approximation of the \textbf{population mean} determined experimentally.
+
+\[ \overline{x} = \dfrac{\Sigma x}{n} \]
+
+where
+\begin{description}[nosep, labelindent=0.5cm]
+  \item \(n\) is the size of the sample (number of sample points)
+  \item \(x\) is the value of a sample point
+\end{description}
+
+\begin{cas}
+  \begin{enumerate}[leftmargin=3mm]
+    \item Spreadsheet
+    \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
+    \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
+    \item Input range as A1:An where \(n\) is the number of samples
+    \item Graph \(\rightarrow\) Histogram
+  \end{enumerate}
+\end{cas}
+
+\subsubsection*{Sample size of \(n\)}
 
-  \begin{cas}
-    \begin{enumerate}[leftmargin=3mm]
-      \item Spreadsheet
-      \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
-      \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
-      \item Input range as A1:An where \(n\) is the number of samples
-      \item Graph \(\rightarrow\) Histogram
-    \end{enumerate}
-  \end{cas}
-
-  \subsubsection*{Sample size of \(n\)}
-
-  \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
-
-  Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
-
-  For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
-
-  \begin{cas}
-
-    \begin{itemize}
-      \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
-      \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
-    \end{itemize}
-
-  \end{cas}
-
-  \subsection*{Normal distributions}
-
-
-  \[ Z = \frac{X - \mu}{\sigma} \]
-
-  Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
-  \(\text{mean} = \text{mode} = \text{median}\)
-
-  \begin{warning}
-    Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
-  \end{warning}
-
-  \pgfmathdeclarefunction{gauss}{2}{%
-    \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
-  }
-  \pgfkeys{/pgf/decoration/.cd,
-    distance/.initial=10pt
-  }
-  \pgfdeclaredecoration{add dim}{final}{
-    \state{final}{% 
-      \pgfmathsetmacro{\dist}{5pt*\pgfkeysvalueof{/pgf/decoration/distance}/abs(\pgfkeysvalueof{/pgf/decoration/distance})}    
-      \pgfpathmoveto{\pgfpoint{0pt}{0pt}}             
-      \pgfpathlineto{\pgfpoint{0pt}{2*\dist}}   
-      \pgfpathmoveto{\pgfpoint{\pgfdecoratedpathlength}{0pt}} 
-      \pgfpathlineto{\pgfpoint{(\pgfdecoratedpathlength}{2*\dist}}     
-      \pgfsetarrowsstart{latex}
-      \pgfsetarrowsend{latex}
-      \pgfpathmoveto{\pgfpoint{0pt}{\dist}}
-      \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{\dist}} 
-      \pgfusepath{stroke} 
-      \pgfpathmoveto{\pgfpoint{0pt}{0pt}}
-      \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{0pt}}
-    }}
-    \tikzset{dim/.style args={#1,#2}{decoration={add dim,distance=#2},
-      decorate,
-      postaction={decorate,decoration={text along path,
-      raise=#2,
-      text align={align=center},
-      text={#1}}}}
-    }
-    \begin{figure*}[hb]
-      \centering
-      \begin{tikzpicture}
-        \begin{axis}[every axis plot post/.style={
-            mark=none,domain=-3:3,samples=50,smooth}, 
-          axis x line=bottom, 
-          axis y line=left,
-          enlargelimits=upper,
-          x=\textwidth/10,
-          ytick={0.55},
-          yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)}, 
-          xtick={-2,-1,0,1,2},
-          x tick label style = {font=\footnotesize},
-          xticklabels={\((\mu-2\sigma)\), \((\mu-\sigma)\), \(\mu\), \((\mu+\sigma)\), \((\mu+2\sigma)\)},
-          xlabel={\(x\)},
-          every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
-          every axis y label/.style={at={(axis description cs:-0.02,0.2)}, anchor=south west, rotate=90},
-          ylabel={\(\Pr(X=x)\)}]
-          \addplot {gauss(0,0.75)};
-          \fill[red!30] (-3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-((x)^2)/(2*0.75^2))} -- (3,0) -- cycle;
-          \fill[darkgray!30] (3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (3,0) -- cycle;
-          \fill[lightgray!30] (-2,0)  -- plot[id=f3,domain=-2:2,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (2,0) -- cycle;
-          \fill[white!30] (-1,0)  -- plot[id=f3,domain=-1:1,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (1,0) -- cycle;
-          \begin{scope}[<->]
-            \draw (-1,0.35) -- (1,0.35) node [midway, fill=white] {68.3\%};
-            \draw (-2,0.25) -- (2,0.25) node [midway, fill=white] {95.5\%};
-            \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.7\%};
-          \end{scope}
-          \begin{scope}[-, dashed, gray]
-            \draw (-1,0) -- (-1, 0.35);
-            \draw (1,0) -- (1, 0.35);
-            \draw (-2,0) -- (-2, 0.25);
-            \draw (2,0) -- (2, 0.25);
-            \draw (-3,0) -- (-3, 0.15);
-            \draw (3,0) -- (3, 0.15);
-          \end{scope}
-        \end{axis}
-        \begin{axis}[every axis plot post/.append style={
-            mark=none,domain=-3:3,samples=50,smooth}, 
-          axis x line=bottom, 
-          enlargelimits=upper,
-          x=\textwidth/10,
-          xtick={-2,-1,0,1,2},
-          axis x line shift=30pt,
-          hide y axis,
-          x tick label style = {font=\footnotesize},
-          xlabel={\(Z\)},
-          every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
-          \addplot {gauss(0,0.75)};
-        \end{axis}
-      \end{tikzpicture}
-    \end{figure*}
-
-  \subsection*{Confidence intervals}
+\[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
+
+Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
+
+For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
+
+\begin{cas}
 
   \begin{itemize}
-    \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
-    \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
-    \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)
+    \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
+    \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
   \end{itemize}
 
-  \subsubsection*{95\% confidence interval}
+\end{cas}
 
-  For 95\% c.i. of population mean \(\mu\):
+\subsection*{Normal distributions}
 
-  \[ x \in \left(\overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \right)\]
 
-  where:
-    \begin{description}[nosep, labelindent=0.5cm]
-    \item \(\overline{x}\) is the sample mean
-    \item \(\sigma\) is the population sd
-    \item \(n\) is the sample size from which \(\overline{x}\) was calculated
-  \end{description}
+\[ Z = \frac{X - \mu}{\sigma} \]
+
+Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
+\(\text{mean} = \text{mode} = \text{median}\)
+
+\begin{warning}
+  Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
+\end{warning}
+
+\subsection*{Confidence intervals}
+
+\begin{itemize}
+  \item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\)
+  \item \textbf{Interval estimate:} confidence interval for population mean \(\mu\)
+  \item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\)
+\end{itemize}
+
+\subsubsection*{95\% confidence interval}
+
+For 95\% c.i. of population mean \(\mu\):
+
+\[ x \in \left(\overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \right)\]
+
+where:
+\begin{description}[nosep, labelindent=0.5cm]
+  \item \(\overline{x}\) is the sample mean
+  \item \(\sigma\) is the population sd
+  \item \(n\) is the sample size from which \(\overline{x}\) was calculated
+\end{description}
 
-  \begin{cas}
-    Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
-    Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
-  \end{cas}
+\begin{cas}
+  Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
+  Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable}
+\end{cas}
 
-  \subsection*{Margin of error}
+\subsection*{Margin of error}
 
-  For 95\% confidence interval of \(\mu\):
-  \begin{align*}
-    M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
-    &= \dfrac{1}{2} \times \text{width of c.i.} \\
-    \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
-  \end{align*}
+For 95\% confidence interval of \(\mu\):
+\begin{align*}
+  M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
+  &= \dfrac{1}{2} \times \text{width of c.i.} \\
+  \implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
+\end{align*}
 
-  Always round \(n\) up to a whole number of samples.
+Always round \(n\) up to a whole number of samples.
 
-  \subsection*{General case}
+\subsection*{General case}
 
-  For \(C\)\% c.i. of population mean \(\mu\):
+For \(C\)\% c.i. of population mean \(\mu\):
 
-  \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \]
-  \hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)
+\[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \]
+\hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)
 
-  \begin{cas}
-    Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
-    Set \textit{Type = One-\colorbox{important}{Prop} Z Int} \\
-    Input  x \(= \hat{p} * n\)
-  \end{cas}
+\begin{cas}
+  Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
+  Set \textit{Type = One-\colorbox{important}{Prop} Z Int} \\
+  Input  x \(= \hat{p} * n\)
+\end{cas}
 
-  \subsection*{Confidence interval for multiple trials}
+\subsection*{Confidence interval for multiple trials}
 
-  For a set of \(n\) confidence intervals (samples), there is \(0.95^n\) chance that all \(n\) intervals contain the population mean \(\mu\).
+For a set of \(n\) confidence intervals (samples), there is \(0.95^n\) chance that all \(n\) intervals contain the population mean \(\mu\).
 
-  \end{document}
+\end{document}
diff --git a/methods/statistics.pdf b/methods/statistics.pdf
index 7a81019..27e441e 100644
Binary files a/methods/statistics.pdf and b/methods/statistics.pdf differ
diff --git a/methods/statistics.tex b/methods/statistics.tex
index 3a205e5..b25d045 100644
--- a/methods/statistics.tex
+++ b/methods/statistics.tex
@@ -203,6 +203,6 @@
 
   \[ \operatorname{E}(X) = \int^b_a x f(x) \> dx \]
 
-  \[ \operatorname{sd}(X) = \sqrt{\operatorname{Var}(X)} = \sqrt{\oepratorname{E}(X^2)-[\operatorname{E}(X)]^2} \]
+  \[ \operatorname{sd}(X) = \sqrt{\operatorname{Var}(X)} = \sqrt{\operatorname{E}(X^2)-[\operatorname{E}(X)]^2} \]
 
 \end{document}