[chem] start organic reactions summary document
[notes.git] / methods / methods-collated.tex
index 2257b41a661dc8927aa8ad8385b60603b8eb52b2..a3a1e796dc6cd2ae5b1aa0b637147ffbc0a1a394 100644 (file)
@@ -1,24 +1,26 @@
-\documentclass[a4paper, twocolumn]{article}
+\documentclass[a4paper]{article}
 \usepackage[dvipsnames, table]{xcolor}
 \usepackage{adjustbox}
 \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}
 \usepackage{pgfplots}
 \usepackage{pst-plot}
 \usepackage{standalone}
+\usepackage{subfiles}
 \usepackage{tabularx}
 \usepackage{tabu}
 \usepackage{tcolorbox}
 
 \usetikzlibrary{%
   angles,
+  arrows,
+  arrows.meta,
   calc,
   datavisualization.formats.functions,
   decorations,
   decorations.markings,
+  decorations.text,
   decorations.pathreplacing,
   decorations.text,
+  patterns,
   scopes
 }
+
 \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+
 \usepgflibrary{arrows.meta}
-\pgfplotsset{compat=1.8}
+\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)
         \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}
 
 \definecolor{cas}{HTML}{e6f0fe}
 \definecolor{important}{HTML}{fc9871}
+\definecolor{highlight}{HTML}{ffb84d}
 \definecolor{dark-gray}{gray}{0.2}
-\definecolor{shade1}{HTML}{ffffff}
-\definecolor{shade2}{HTML}{e6f2ff}
-\definecolor{shade3}{HTML}{cce2ff}
+\definecolor{peach}{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}
 \date{}
 \maketitle
 
+\begin{multicols}{2}
+
 
 \section{Functions}
 
@@ -365,281 +377,8 @@ For \(x^n\), parity of \(n \equiv\) parity of function
       \input{circ-functions}
       \input{calculus}
 
+      \subfile{statistics-ref}
 
+    \end{multicols}
 
-      \section{Statistics}
-
-      \subsection*{Probability}
-
-      \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*}
-
-      Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
-
-      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*}
-
-      \subsection*{Combinatorics}
-
-      \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*{Distributions}
-
-      \subsubsection*{Mean \(\mu\)}
-
-      \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
-
-      \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*}
-
-      \subsubsection*{Mode}
-
-      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*{Median}
-
-      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.
-
-      \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
-
-      \subsubsection*{Variance \(\sigma^2\)}
-
-      \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*{Standard deviation \(\sigma\)}
-
-      \begin{align*}
-        \sigma &= \operatorname{sd}(X) \\
-        &= \sqrt{\operatorname{Var}(X)}
-      \end{align*}
-
-      \subsection*{Binomial distributions}
-
-      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}
-
-
-      \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*}
-
-      \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}
-
-      \subsection*{Continuous random variables}
-
-      A continuous random variable \(X\) has a pdf \(f\) such that:
-
-      \begin{enumerate}
-        \item \(f(x) \ge 0 \forall x \)
-        \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
-      \end{enumerate}
-
-      \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*{Linear functions \(X \rightarrow aX+b\)}
-
-      \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*}
-
-      \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*}
-
-      \subsection*{Expectation theorems}
-
-      For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
-
-      \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*}
-
-      \subsection*{Sample mean}
-
-      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 \(n\)}
-
-      \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]
-
-      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*}
-          \end{document}
+\end{document}