-\documentclass[a4paper]{article}
-\usepackage{standalone}
-\usepackage{newclude}
-\usepackage[a4paper,margin=2cm]{geometry}
-\usepackage{multicol}
-\usepackage{multirow}
+\documentclass[a4paper, twocolumn]{article}
+\usepackage[dvipsnames, table]{xcolor}
+\usepackage{adjustbox}
\usepackage{amsmath}
\usepackage{amssymb}
+\usepackage{blindtext}
+\usepackage{enumitem}
+\usepackage{fancyhdr}
+\usepackage[a4paper,margin=2cm]{geometry}
+\usepackage{graphicx}
\usepackage{harpoon}
+\usepackage{listings}
+\usepackage{longtable}
+\usepackage{makecell}
+\usepackage{mathtools}
+\usepackage{multicol}
+\usepackage{multirow}
+\usepackage{newclude}
+\usepackage{pgfplots}
+\usepackage{pst-plot}
+\usepackage{standalone}
\usepackage{tabularx}
\usepackage{tabu}
-\usepackage{makecell}
-\usepackage[dvipsnames, table]{xcolor}
-\usepackage{blindtext}
-\usepackage{graphicx}
-\usepackage{wrapfig}
-\usepackage{tikz}
+\usepackage{tcolorbox}
\usepackage{tikz-3dplot}
-\usepackage{pgfplots}
-\pgfplotsset{compat=1.8}
-\usepackage{mathtools}
-\usetikzlibrary{calc}
-\usetikzlibrary{angles}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
+\usepackage{tikz}
+\usepackage{tkz-fct}
+\usepackage[obeyspaces]{url}
+\usepackage{wrapfig}
+
+
+\usetikzlibrary{%
+ angles,
+ calc,
+ datavisualization.formats.functions,
+ decorations,
+ decorations.markings,
+ decorations.pathreplacing,
+ decorations.text,
+ scopes
+}
+\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
\usepgflibrary{arrows.meta}
-\usepackage{longtable}
-\usepackage{fancyhdr}
+\pgfplotsset{compat=1.8}
+\psset{dimen=monkey,fillstyle=solid,opacity=.5}
+\def\object{%
+ \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
+ \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90]
+ \rput{*0}{%
+ \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
+
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\linespread{1.5}
\setlength{\parindent}{0cm}
-\usepackage{mathtools}
-\usepackage{xcolor} % used only to show the phantomed stuff
\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}
+\newcolumntype{Y}{>{\centering\arraybackslash}X}
+
\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}
-\linespread{1.5}
-\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
-\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={$y$},
-}}
+
+\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}
+
\begin{document}
\date{}
\maketitle
-\begin{multicols}{2}
-
- \section{Functions}
-
- \begin{itemize}
- \tightlist
- \item vertical line test
- \item each \(x\) value produces only one \(y\) value
- \end{itemize}
- \subsection*{One to one functions}
+\section{Functions}
- \begin{itemize} \tightlist
- \item
- \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
- \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
- \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
- \(x^3\) is)
- \item
- horizontal line test
- \item
- if not one to one, it is many to one
- \end{itemize}
+\begin{itemize} \tightlist
+ \item vertical line test
+ \item each \(x\) value produces only one \(y\) value
+\end{itemize}
- \subsection*{Odd and even functions}
+\subsection*{One to one functions}
+
+\begin{itemize} \tightlist
+ \item
+ \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
+ \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
+ \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
+ \(x^3\) is)
+ \item
+ horizontal line test
+ \item
+ if not one to one, it is many to one
+\end{itemize}
- \begin{align*}
- \text{Even:}&& f(x) &= f(-x) \\
- \text{Odd:} && -f(x) &= f(-x)
- \end{align*}
+\subsection*{Odd and even functions}
+
+\begin{align*}
+ \text{Even:}&& f(x) &= f(-x) \\
+ \text{Odd:} && -f(x) &= f(-x)
+\end{align*}
+
+Even \(\implies\) symmetrical across \(y\)-axis \\
+\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
+For \(x^n\), parity of \(n \equiv\) parity of function
+
+\begin{tabularx}{\columnwidth}{XX}
+ \textbf{Even:} & \textbf{Odd:} \\
+ \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} &
+ \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)}; \end{axis}\end{tikzpicture}
+\end{tabularx}
+
+\subsection*{Inverse functions}
+
+\begin{itemize} \tightlist
+ \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
+ \item \(f\) must be one to one
+ \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
+ \item Represents reflection across \(y=x\)
+ \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
+ \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
+ \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
+ \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
+\end{itemize}
- Even \(\implies\) symmetrical across \(y\)-axis \\
- \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
- For \(x^n\), parity of \(n \equiv\) parity of function
+\subsubsection*{Finding \(f^{-1}\)}
- \begin{tabularx}{\columnwidth}{XX}
- \textbf{Even:} & \textbf{Odd:} \\
- \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)}; \end{axis}\end{tikzpicture}
- \end{tabularx}
+\begin{enumerate} \tightlist
+ \item Let \(y=f(x)\)
+ \item Swap \(x\) and \(y\) (``take inverse''
+ \item Solve for \(y\) \\
+ Sqrt: state \(\pm\) solutions then restrict
+ \item State rule as \(f^{-1}(x)=\dots\)
+ \item For inverse \emph{function}, state in function notation
+\end{enumerate}
- \subsection*{Inverse functions}
+\subsection*{Simultaneous equations (linear)}
- \begin{itemize} \tightlist
- \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
- \item \(f\) must be one to one
- \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
- \item Represents reflection across \(y=x\)
- \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
- \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
- \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
- \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
- \end{itemize}
+\begin{itemize} \tightlist
+ \item \textbf{Unique solution} - lines intersect at point
+ \item \textbf{Infinitely many solutions} - lines are equal
+ \item \textbf{No solution} - lines are parallel
+\end{itemize}
- \subsubsection*{Finding \(f^{-1}\)}
+\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
+ where all coefficients are known except for one, and \(a, b\) are known
\begin{enumerate} \tightlist
- \item Let \(y=f(x)\)
- \item Swap \(x\) and \(y\) (``take inverse''
- \item Solve for \(y\) \\
- Sqrt: state \(\pm\) solutions then restrict
- \item State rule as \(f^{-1}(x)=\dots\)
- \item For inverse \emph{function}, state in function notation
+ \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
+ \item Find determinant of first matrix: \(\Delta = ps-qr\)
+ \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
+ or let \(\Delta \ne 0\) for one unique solution.
+ \item Solve determinant equation to find variable \\
+ \textbf{For infinite/no solutions:}
+ \item Substitute variable into both original equations
+ \item Rearrange equations so that LHS of each is the same
+ \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
+ \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
\end{enumerate}
-
- \subsection*{Simultaneous equations (linear)}
-
- \begin{itemize} \tightlist
- \item \textbf{Unique solution} - lines intersect at point
- \item \textbf{Infinitely many solutions} - lines are equal
- \item \textbf{No solution} - lines are parallel
- \end{itemize}
-
- \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
- where all coefficients are known except for one, and \(a, b\) are known
-
- \begin{enumerate} \tightlist
- \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
- \item Find determinant of first matrix: \(\Delta = ps-qr\)
- \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
- or let \(\Delta \ne 0\) for one unique solution.
- \item Solve determinant equation to find variable \\
- \textbf{For infinite/no solutions:}
- \item Substitute variable into both original equations
- \item Rearrange equations so that LHS of each is the same
- \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
- \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
- \end{enumerate}
-
- \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
-
- \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
-
- \begin{itemize} \tightlist
- \item Use elimination
- \item Generate two new equations with only two variables
- \item Rearrange \& solve
- \item Substitute one variable into another equation to find another variable
- \end{itemize}
-\subsection*{Piecewise functions}
+ \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
-\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
+ \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
-\textbf{Open circle:} point included\\
-\textbf{Closed circle:} point not included
+ \begin{itemize} \tightlist
+ \item Use elimination
+ \item Generate two new equations with only two variables
+ \item Rearrange \& solve
+ \item Substitute one variable into another equation to find another variable
+ \end{itemize}
-\subsection*{Operations on functions}
+ \subsection*{Piecewise functions}
-For \(f \pm g\) and \(f \times g\):
-\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
+ \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
-Addition of linear piecewise graphs: add \(y\)-values at key points
+ \textbf{Open circle:} point included\\
+ \textbf{Closed circle:} point not included
-Product functions:
+ \subsection*{Operations on functions}
-\begin{itemize}
-\tightlist
-\item
- product will equal 0 if \(f=0\) or \(g=0\)
-\item
- \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
-\end{itemize}
+ For \(f \pm g\) and \(f \times g\):
+ \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
-\subsection*{Composite functions}
+ Addition of linear piecewise graphs: add \(y\)-values at key points
-\((f \circ g)(x)\) is defined iff
-\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
+ Product functions:
+ \begin{itemize}
+ \tightlist
+ \item
+ product will equal 0 if \(f=0\) or \(g=0\)
+ \item
+ \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
+ \end{itemize}
- \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks
- \begin{table*}[ht]
+ \subsection*{Composite functions}
+
+ \((f \circ g)(x)\) is defined iff
+ \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
+
+ \pgfplotsset{
+ blank/.append style={%
+ enlargelimits=true,
+ ticks=none,
+ yticklabels={,,}, xticklabels={,,},
+ xlabel=, ylabel=,
+ scale=0.4,
+ samples=100, smooth, unbounded coords=jump
+ }
+ }
+ \tikzset{
+ blankplot/.append style={orange, mark=none}
+ }
+
+ \begin{figure*}[ht]
\centering
- \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}}
- & \(n\) is even & \(n\) is odd \\ \tabucline{1pt}
- \(x^n, n \in \mathbb{Z}^+\) &
- \vspace{1em}\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)}; \end{axis}\end{tikzpicture} \\
- \(x^n, n \in \mathbb{Z}^-\) &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4, xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none, domain=-3:-0.1] {(x^(-1))}; \addplot[orange, mark=none, domain=0.1:3] {(x^(-1))}; \end{axis}\end{tikzpicture} \\
- \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
- \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1, xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))}; \end{axis}\end{tikzpicture} &
- \begin{tikzpicture}
- \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]
- \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
- \end{axis}
- \end{tikzpicture}
- \end{tabu}
- \hrule
- \end{table*}
- \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back
+
+ \begin{tabularx}{\textwidth}{r|Y|Y}
+
+ & \(n\) is even & \(n\) is odd \\ \hline
+
+ \centering \(x^n, n \in \mathbb{Z}^+\) &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3]
+ \addplot[blankplot] {(x^2)};
+ \end{axis}
+ \end{tikzpicture}} &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3]
+ \addplot[blankplot, domain=-3:3] {(x^3)};
+ \end{axis}
+ \end{tikzpicture}} \\ \hline
+
+ \centering \(x^n, n \in \mathbb{Z}^-\) &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
+ \addplot[blankplot, samples=100] {(x^(-2))};
+ \end{axis}
+ \end{tikzpicture}} &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3]
+ \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
+ \addplot[blankplot, domain=0.1:3] {(x^(-1))};
+ \end{axis}
+ \end{tikzpicture}} \\ \hline
+
+ \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-1, xmax=5]
+ \addplot[blankplot] {(x^(1/2))};
+ \end{axis}
+ \end{tikzpicture}} &
+
+ \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
+ \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
+ \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
+ \end{axis}
+ \end{tikzpicture}} \\ \hline
+
+ \end{tabularx}
+ \end{figure*}
\section{Polynomials}
\input{circ-functions}
\input{calculus}
- \end{multicols}
- \end{document}
+
+
+ \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}