-\documentclass[a4paper, tikz, pstricks]{article}
-\usepackage[a4paper,margin=2cm]{geometry}
-\usepackage{array}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{tcolorbox}
-\usepackage{fancyhdr}
-\usepackage{pgfplots}
-\usepackage{tikz}
-\usetikzlibrary{arrows,
- calc,
- decorations,
- scopes,
- angles
-}
-\usetikzlibrary{calc}
-\usetikzlibrary{angles}
-\usetikzlibrary{datavisualization.formats.functions}
-\usetikzlibrary{decorations.markings}
-\usepgflibrary{arrows.meta}
-\usetikzlibrary{decorations.markings}
-\usepgflibrary{arrows.meta}
-\usepackage{pst-plot}
-\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){$\vec{w}$}}
-}
-
-\usepackage{tabularx}
-\usetikzlibrary{angles}
-\usepackage{keystroke}
-\usepackage{listings}
-\usepackage{xcolor} % used only to show the phantomed stuff
-\definecolor{cas}{HTML}{e6f0fe}
-
-\pagestyle{fancy}
-\fancyhead[LO,LE]{Year 12 Specialist - Dynamics}
-\fancyhead[CO,CE]{Andrew Lorimer}
-
-\setlength\parindent{0pt}
-
+\documentclass[spec-collated.tex]{subfiles}
\begin{document}
-\title{Dynamics}
-\author{}
-\date{}
-\maketitle
+\section{Dynamics}
-\section{Resolution of forces}
+\subsection*{Resolution of forces}
\textbf{Resultant force} is sum of force vectors
-\subsection*{In angle-magnitude form}
+\subsubsection*{In angle-magnitude form}
\makebox[3cm]{Cosine rule:} \(c^2=a^2+b^2-2ab\cos\theta\)
\makebox[3cm]{Sine rule:} \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\)
-\subsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
+\subsubsection*{In \(\boldsymbol{i}\)---\(\boldsymbol{j}\) form}
Vector of \(a\) N at \(\theta\) to \(x\) axis is equal to \(a \cos \theta \boldsymbol{i} + a \sin \theta \boldsymbol{j}\). Convert all force vectors then add.
To find angle of an \(a\boldsymbol{i} + b\boldsymbol{j}\) vector, use \(\theta = \tan^{-1} \frac{b}{a}\)
-\subsection*{Resolving in a given direction}
+\subsubsection*{Resolving in a given direction}
The resolved part of a force \(P\) at angle \(\theta\) is has magnitude \(P \cos \theta\)
To convert force \(||\vec{OA}\) to angle-magnitude form, find component \(\perp\vec{OA}\) then \(|\boldsymbol{r}|=\sqrt{\left(||\vec{OA}\right)^2 + \left(\perp\vec{OA}\right)^2},\quad \theta = \tan^{-1}\dfrac{\perp\vec{OA}}{||\vec{OA}}\)
-\section{Newton's laws}
+\subsection*{Newton's laws}
\begin{tcolorbox}
- \begin{enumerate}
- \item Velocity is constant without a net external velocity
+ \begin{enumerate}[leftmargin=1mm]
+ \item Velocity is constant without a net external force
\item \(\frac{d}{dt} \rho \propto \Sigma F \implies \boldsymbol{F}=m\boldsymbol{a}\)
\item Equal and opposite forces
\end{enumerate}
\end{tcolorbox}
-\subsection*{Weight}
+\subsubsection*{Weight}
A mass of \(m\) kg has force of \(mg\) acting on it
-\subsection*{Momentum \(\rho\)}
+\subsubsection*{Momentum \(\rho\)}
\[ \rho = mv \tag{units kg m/s or Ns} \]
-\subsection*{Reaction force \(R\)}
+\subsubsection*{Reaction force \(R\)}
\begin{itemize}
\item With no vertical velocity, \(R=mg\)
- \item With upwards acceleration, \(R-mg=ma\)
+ \item With vertical acceleration, \(|R|=m|a|-mg\)
\item With force \(F\) at angle \(\theta\), then \(R=mg-F\sin\theta\)
\end{itemize}
-\subsection*{Friction}
+\subsubsection*{Friction}
\[ F_R = \mu R \tag{friction coefficient} \]
-\section{Inclined planes}
+\subsection*{Inclined planes}
\[ \boldsymbol{F} = |\boldsymbol{F}| \cos \theta \boldsymbol{i} + |\boldsymbol{F}| \sin \theta \boldsymbol{j} \]
\begin{itemize}
pulley/.style={thick}
}
-\begin{figure}[!htb]
- \centering
- \begin{tikzpicture}
+ \begin{center}\begin{tikzpicture}
\pgfmathsetmacro{\Fnorme}{2}
\pgfmathsetmacro{\Fangle}{30}
\end{scope}
\draw[force,->] (M.center) -- ++(0,-1) node[below] {$mg$};
\draw (M.center)+(-90:\arcr) arc [start angle=-90,end angle=\iangle-90,radius=\arcr] node [below, pos=.5] {\footnotesize\(\theta\)};
- \end{tikzpicture}
-\end{figure}
+ \end{tikzpicture}\end{center}
-\section{Connected particles}
-
-\begin{itemize}
- \item \textbf{Suspended pulley:} tension in both sections of rope are equal
- \item \textbf{Linear connection:} find acceleration of system first
- \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
- \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
-\end{itemize}
+\subsection*{Connected particles}
\def\boxwidth{0.5}
\tikzset{
}
-\begin{figure}[!htb]
- \centering
+\begin{center}
\begin{tikzpicture}
\matrix[column sep=1cm] {
\\
};
\end{tikzpicture}
-\end{figure}
+ \end{center}
+
+\begin{itemize}
+ \item \textbf{Suspended pulley:} tension in both sections of rope are equal \\
+ \(|a| = g \frac{m_1 - m_2}{m_1 + m_2}\) where \(m_1\) accelerates down \\
+ With tension: \\
+ \[ \begin{cases}m_1 g - T = m_1 a\\ T - m_2 g = m_2 a\end{cases} \\ \implies m_1 g - m_2 g = m_1 a + m_2 a \]
+ \item \textbf{String pulling mass on inclined pane:} Resolve parallel to plane \\
+ \[ T-mg \sin \theta = ma \]
+ \item \textbf{Linear connection:} find acceleration of system first
+ \item \textbf{Pulley on right angle:} \(a = \frac{m_2g}{m_1+m_2}\) where \(m_2\) is suspended (frictionless on both surfaces)
+ \item \textbf{Pulley on edge of incline:} find downwards force \(W_2\) and components of mass on plane
+\end{itemize}
+
+\hspace{2em}\parbox{8em}{In this example, note \(T_1 \ne T_2\):}
+ \begin{tikzpicture}
+
+ \begin{scope}
-\section{Equilibrium}
+ \coordinate (O) at (0,0);
+ \coordinate (A) at ($({3*cos(\iangle)},{3*sin(\iangle)})$);
+ \coordinate (B) at ($({3*cos(\iangle)},0)$);
+ \coordinate (C) at ($({(1-0.25*\boxwidth)*cos(\iangle)},{(1-0.25*\boxwidth)*sin(\iangle)})$); % centre of box
+ \coordinate (D) at ($(C)+(\iangle:\boxwidth)$);
+ \coordinate (E) at ($(D)+(90+\iangle:0.5*\boxwidth)$);
+ \coordinate (F) at ($(B)+(0,{1.5*sin(\iangle)})$);
+ \coordinate (G) at ($(A)+(\iangle:-2*\boxwidth)$);
+ \coordinate (H) at ($(G)+(90+\iangle:0.5*\boxwidth)$);
+ \coordinate (I) at ($(H)+(\iangle:-0.5*\boxwidth)$);
+ \coordinate (J) at ($(H)+(\iangle:\boxwidth)$);
+ \coordinate (X) at ($(A)+(\iangle:0.5*\boxwidth)$); % centre of pulley
+ \coordinate (Y) at ($(X)+(90+\iangle:0.5*\boxwidth)$); % chord of pulley
+
+ \draw[plane] (O) -- (A) -- (B) -- (O);
+ \draw (O)+(\arcr,0) arc [start angle=0,end angle=\iangle,radius=\arcr] node [right, pos=.75] {\footnotesize\(\theta\)};
+
+ \draw [rotate=\iangle, m] (C) rectangle ++(\boxwidth,\boxwidth) node (z) [rotate=\iangle, midway, font=\footnotesize] {\(m_1\)};
+ \draw [rotate=\iangle, m] (G) rectangle ++(\boxwidth,\boxwidth) node (l) [rotate=\iangle, midway, font=\footnotesize] {\(m_2\)};
+ \draw [pulley] (A) -- (X) ++(0.5*\boxwidth, 0) arc[rotate=\iangle, start angle=0, delta angle=360, x radius=0.25, y radius=0.25] node(r) [midway, rotate=\iangle] {};
+ \draw [string] (E) -- (H) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_2\)};
+ \draw [string] (J) -- (Y) node [midway, above, font=\footnotesize, rotate=\iangle] {\(T_1\)} arc (90+\iangle:0:0.25) -- ++($(0,{-1.5*sin(\iangle)})$) node [midway, above right, font=\footnotesize] {\(T_1\)} node[m] {\(m_3\)};
+
+ \end{scope}
+
+ \end{tikzpicture}
+\subsection*{Equilibrium}
\[ \dfrac{A}{\sin a} = \dfrac{B}{\sin b} = \dfrac{C}{\sin c} \tag{Lami's theorem}\]
+\[ c^2 = a^2 + b^2 - 2ab \cos \theta \tag{cosine rule} \]
Three methods:
\begin{enumerate}
\item Lami's theorem (sine rule)
- \item Triangle of forces or CAS (use to verify)
+ \item Triangle of forces (cosine rule)
\item Resolution of forces (\(\Sigma F = 0\) - simultaneous)
\end{enumerate}
-\colorbox{cas}{On CAS:} use Geometry, lock known constants.
+ \begin{cas}
+ \textbf{To verify:} Geometry tab, then select points with normal cursor. Click right arrow at end of toolbar and input point, then lock known constants.
+ \end{cas}
-\section{Variable forces (DEs)}
+\subsection*{Variable forces (DEs)}
\[ a = \dfrac{d^2x}{dt^2} = \dfrac{dv}{dt} = v\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \frac{1}{2} v^2 \right) \]
\documentclass[a4paper]{article}
\usepackage[a4paper,margin=2cm]{geometry}
\usepackage{multicol}
+\usepackage{dblfloatfix}
\usepackage{multirow}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{harpoon}
\usepackage{tabularx}
\usepackage{makecell}
+\usepackage{enumitem}
+\usepackage[obeyspaces]{url}
\usepackage[dvipsnames, table]{xcolor}
\usepackage{blindtext}
\usepackage{graphicx}
\usepackage{tkz-fct}
\usepackage{tikz-3dplot}
\usepackage{pgfplots}
+\usetikzlibrary{arrows,
+ decorations,
+ decorations.markings,
+ decorations.text,
+ scopes
+}
+\usetikzlibrary{datavisualization.formats.functions}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usepackage{pst-plot}
+\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){$\vec{w}$}}
+}
+
\usetikzlibrary{calc}
\usetikzlibrary{angles}
\usetikzlibrary{datavisualization.formats.functions}
\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!
\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
- }}
+\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}
\(f^{\prime\prime} = 0\))
- \pgfplotsset{every axis/.append style={
- axis x line=none, % put the x axis in the middle
- axis y line=none, % put the y axis in the middle
- }}
\begin{table*}[ht]
\centering
\begin{tabularx}{\textwidth}{rXXX}
& \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
\hline
\(\dfrac{dy}{dx}>0\) &
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
\hline
\(\dfrac{dy}{dx}<0\) &
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0, xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)}; \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=1.5, xmax=4.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0, xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)}; \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5, xmax=4.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
\hline
\(\dfrac{dy}{dx}=0\)&
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
- \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
+ \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\
\hline
\end{tabularx}
\end{table*}
Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
\[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
+ \subfile{dynamics}
+ \subfile{statistics}
\end{multicols}
\end{document}
-\documentclass[a4paper]{article}
-\usepackage[a4paper, margin=2cm]{geometry}
-\usepackage{array}
-\usepackage{amsmath}
-\usepackage{amssymb}
-\usepackage{tcolorbox}
-\usepackage{fancyhdr}
-\usepackage{pgfplots}
-\usepackage{tabularx}
-\usepackage{keystroke}
-\usepackage{listings}
-\usepackage{xcolor} % used only to show the phantomed stuff
-\definecolor{cas}{HTML}{e6f0fe}
-\usepackage{mathtools}
-\pgfplotsset{compat=1.16}
-
-\pagestyle{fancy}
-\fancyhead[LO,LE]{Unit 4 Specialist --- Statistics}
-\fancyhead[CO,CE]{Andrew Lorimer}
-
-\setlength\parindent{0pt}
-
+\documentclass[spec-collated.tex]{subfiles}
\begin{document}
- \title{Statistics}
- \author{}
- \date{}
- \maketitle
-
- \section{Linear combinations of random variables}
+ \section{Statistics}
\subsection*{Continuous random variables}
\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*}
+
\[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
+
- \subsubsection*{Linear functions \(X \rightarrow aX+b\)}
+ \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) \\
\textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
\end{align*}
- \subsection*{Linear combination of two random variables}
+ \subsection*{Expectation theorems}
+
+ For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
\begin{align*}
- \textbf{Mean:} && \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
- \textbf{Variance:} && \operatorname{Var}(aX+bY) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \tag{if \(X\) and \(Y\) are independent}\\
+ 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*}
- \section{Sample mean}
+ \subsection*{Sample mean}
Approximation of the \textbf{population mean} determined experimentally.
\[ \overline{x} = \dfrac{\Sigma x}{n} \]
- where \(n\) is the size of the sample (number of sample points) and \(x\) is the value of a sample point
-
- \begin{tcolorbox}[colframe=cas!75!black, title=On CAS]
+ 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{enumerate}
+\begin{cas}
+ \begin{enumerate}[leftmargin=3mm]
\item Spreadsheet
- \item In cell A1: \verb;mean(randNorm(sd, mean, sample size));
+ \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{tcolorbox}
+ \end{cas}
\subsubsection*{Sample size of \(n\)}
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{tcolorbox}[colframe=cas!75!black, title=On CAS]
+ \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{tcolorbox}
+
+ \end{cas}
- \section{Normal distributions}
+ \subsection*{Normal distributions}
- mean = mode = median
\[ Z = \frac{X - \mu}{\sigma} \]
- Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\)
+ 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))}
+ \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}
-
-\begin{tikzpicture}
- \pgfplotsset{set layers}
+ \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,
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},
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{tikzpicture}\end{center}}
+ \end{figure*}
- \section{Central limit theorem}
+ \subsection*{Central limit theorem}
If \(X\) is randomly distributed with mean \(\mu\) and sd \(\sigma\), then with an adequate sample size \(n\) the distribution of the sample mean \(\overline{X}\) is approximately normal with mean \(E(\overline{X})\) and \(\operatorname{sd}(\overline{X}) = \frac{\sigma}{\sqrt{n}}\).
- \section{Confidence intervals}
+ \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}
- \subsection{95\% confidence interval}
-
- The 95\% confidence interval for a population mean \(\mu\) is given by
-
- \[ \overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \]
-
- where: \\
- \(\overline{x}\) is the sample mean \\
- \(\sigma\) is the population sd \\
- \(n\) is the sample size from which \(\overline{x}\) was calculated
+ \subsubsection*{95\% confidence interval}
- Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
+ For 95\% c.i. of population mean \(\mu\):
- \colorbox{cas}{\textbf{On CAS}}
+ \[ x \in \left(\overline{x} \pm 1.96 \dfrac{\sigma}{\sqrt{n}} \right)\]
- Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\
- Set Type = One-Sample Z Int, Variable
+ 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}
- \subsection*{Interpretation of confidence intervals}
-
- 95\% confidence interval \(\implies\) 95\% of samples will contain population mean \(\mu\).
+ \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}
- For 95\% confidence interval for \(\mu\), margin of error \(M\) is:
-
+ For 95\% confidence interval of \(\mu\):
\begin{align*}
M &= 1.96 \times \dfrac{\sigma}{\sqrt{n}} \\
\implies n &= \left( \dfrac{1.96 \sigma}{M} \right)^2
\end{align*}
+ Always round \(n\) up to a whole number of samples.
+
\subsection*{General case}
- A confidence interval of \(C\)\% for a mean \(\mu\) s given by
+ For \(C\)\% c.i. of population mean \(\mu\):
- \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \quad \text{ where } k \text{ is such that } \Pr(-k < Z < k) = \frac{C}{100} \]
+ \[ x \in \left( \overline{x} \pm k \dfrac{\sigma}{\sqrt{n}} \right) \]
+ \hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\)
\subsection*{Confidence interval for multiple trials}
\section{Hypothesis testing}
- Note hypotheses are always expressed in terms of population parameters
+ \begin{warning}
+ Note hypotheses are always expressed in terms of population parameters
+ \end{warning}
\subsection*{Null hypothesis \(H_0\)}
Significance level is denoted by \(\alpha\).
- If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
- If \(p>\alpha\), null hypothesis is \textbf{accepted}
+ \-\hspace{1em} If \(p<\alpha\), null hypothesis is \textbf{rejected} \\
+ \-\hspace{1em} If \(p>\alpha\), null hypothesis is \textbf{accepted}
\subsection*{\(z\)-test}
Hypothesis test for a mean of a sample drawn from a normally distributed population with a known standard deviation.
- \subsubsection*{\colorbox{cas}{\textbf{On CAS:}}}
-
+ \begin{cas}
Menu \(\rightarrow\) Statistics \(\rightarrow\) Calc \(\rightarrow\) Test. \\
Select \textit{One-Sample Z-Test} and \textit{Variable}, then input:
- \begin{itemize}
- \item \(\mu\) condition - same operator as \(H_1\)
- \item \(\mu_0\) - expected sample mean (null hypothesis)
- \item \(\sigma\) - standard deviation (null hypothesis)
- \item \(\overline{x}\) - sample mean
- \item \(n\) - sample size
- \end{itemize}
+ \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=2cm, font=\normalfont]
+ \item[\(\mu\) cond:] same operator as \(H_1\)
+ \item[\(\mu_0\):] expected sample mean (null hypothesis)
+ \item[\(\sigma\):] standard deviation (null hypothesis)
+ \item[\(\overline{x}\):] sample mean
+ \item[\(n\):] sample size
+ \end{description}
+ \end{cas}
\end{document}