--- /dev/null
+\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{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}
\documentclass[a4paper]{article}
-\usepackage[a4paper,margin=2cm]{geometry}
-\usepackage{multicol}
-\usepackage{dblfloatfix}
-\usepackage{multirow}
+\usepackage[dvipsnames, table]{xcolor}
\usepackage{amsmath}
\usepackage{amssymb}
-\usepackage{harpoon}
-\usepackage{tabularx}
-\usepackage{makecell}
-\usepackage{enumitem}
-\usepackage[obeyspaces]{url}
-\usepackage[dvipsnames, table]{xcolor}
\usepackage{blindtext}
+\usepackage{dblfloatfix}
+\usepackage{enumitem}
+\usepackage{fancyhdr}
+\usepackage[a4paper,margin=2cm]{geometry}
\usepackage{graphicx}
-\usepackage{wrapfig}
+\usepackage{harpoon}
+\usepackage{import}
+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{makecell}
+\usepackage{mathtools}
+\usepackage{mathtools}
+\usepackage{multicol}
+\usepackage{multirow}
+\usepackage{pgfplots}
+\usepackage{pst-plot}
+\usepackage{subfiles}
+\usepackage{tabularx}
+\usepackage{tcolorbox}
+\usepackage{tikz-3dplot}
\usepackage{tikz}
\usepackage{tkz-fct}
-\usepackage{tikz-3dplot}
-\usepackage{pgfplots}
-\usetikzlibrary{arrows,
- decorations,
- decorations.markings,
- decorations.text,
- decorations.pathreplacing,
- scopes
+\usepackage[obeyspaces]{url}
+\usepackage{wrapfig}
+
+
+\usetikzlibrary{%
+ angles,
+ arrows,
+ arrows.meta,
+ calc,
+ datavisualization.formats.functions,
+ decorations,
+ decorations.markings,
+ decorations.text,
+ decorations.pathreplacing,
+ decorations.text,
+ scopes
}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
-\usepgflibrary{arrows.meta}
-\usetikzlibrary{decorations.markings}
+
+\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
+
\usepgflibrary{arrows.meta}
-\usepackage{pst-plot}
+\pgfplotsset{compat=1.16}
+\pgfplotsset{every axis/.append style={
+ axis x line=middle, % put the x axis in the middle
+ axis y line=middle, % put the y axis in the middle
+ axis line style={->}, % arrows on the axis
+ xlabel={$x$}, % default put x on x-axis
+ ylabel={$y$}, % default put y on y-axis
+}}
+
\psset{dimen=monkey,fillstyle=solid,opacity=.5}
\def\object{%
\psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
\uput[-90]{*0}(0,-2){$\vec{w}$}}
}
-\usetikzlibrary{calc}
-\usetikzlibrary{angles}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
-\usepgflibrary{arrows.meta}
-\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead[LO,LE]{Year 12 Specialist}
\fancyhead[CO,CE]{Andrew Lorimer}
-\usepackage{mathtools}
-\usepackage{xcolor} % used only to show the phantomed stuff
+
\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
-\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
+\linespread{1.5}
+\setlength{\parindent}{0pt}
+\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
+
\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{dark-gray}{gray}{0.2}
-\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, % put the x axis in the middle
- axis y line=middle, % put the y axis in the middle
- axis line style={->}, % arrows on the axis
- xlabel={$x$}, % default put x on x-axis
- ylabel={$y$}, % default put y on y-axis
-}}
-\usepackage{tcolorbox}
+
\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
-\usepackage{keystroke}
-\usepackage{listings}
-\usepackage{mathtools}
-\pgfplotsset{compat=1.16}
-\usepackage{subfiles}
-\usepackage{import}
-\setlength{\parindent}{0pt}
+
\begin{document}
\begin{multicols}{2}
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))}%
-}
- \pgfplotsset{every axis/.append style={
- axis x line=middle, % put the x axis in the middle
- axis y line=middle, % put the y axis in the middle
- }} \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{center} \begin{tikzpicture}
- \pgfplotsset{set layers, axis x line=middle, axis y line=middle}
-\begin{axis}[every axis plot post/.append 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{center}}
- \end{figure*}
+ \begin{figure*}[hb]
+ \centering
+ \include{normal-dist-graph}
+ \end{figure*}
\subsection*{Central limit theorem}