1\documentclass[a4paper]{article} 2\usepackage[a4paper,margin=2cm]{geometry} 3\usepackage{array} 4\usepackage{amsmath} 5\usepackage{amssymb} 6\usepackage{tcolorbox} 7\usepackage{fancyhdr} 8\usepackage{pgfplots} 9\usepackage{tabularx} 10 11\pagestyle{fancy} 12\fancyhead[LO,LE]{Unit 3 Methods Statistics} 13\fancyhead[CO,CE]{Andrew Lorimer} 14 15\setlength\parindent{0pt} 16 17\begin{document} 18 19\title{Statistics} 20\author{} 21\date{} 22\maketitle 23 24\section{Probability} 25 26 \[\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B) \] 27 \[\Pr(A \cup B) = 0\tag{mutually exclusive} \] 28 29\section{Conditional probability} 30 31 \[\Pr(A|B) = \frac{\Pr(A \cap B)}{\Pr(B)}\quad \text{where }\Pr(B) \ne0 \] 32 33 \[\Pr(A) = \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) \tag{law of total probability} \] 34 35 \[\Pr(A \cap B) = \Pr(A|B) \times \Pr(B) \tag{multiplication theorem} \] 36 37 For independent events: 38 39\begin{itemize} 40\item \(\Pr(A \cap B) = \Pr(A) \times \Pr(B)\) 41\item \(\Pr(A|B) = \Pr(A)\) 42\item \(\Pr(B|A) = \Pr(B)\) 43\end{itemize} 44 45\subsection{Discrete random distributions} 46 47 Any experiment or activity involving chance will have a probability associated with each result or \textit{outcome}. If the outcomes have a reference to \textbf{discrete numeric values} (outcomes that can be counted), and the result is unknown, then the activity is a \textit{discrete random probability distribution}. 48 49\subsubsection{Discrete probability distributions} 50 51 If an activity has outcomes whose probability values are all positive and less than one ($\implies0\le p(x) \le1$), and for which the sum of all outcome probabilities is unity ($\implies \sum p(x) = 1$), then it is called a \textit{probability distribution} or \textit{probability mass} function. 52 53\begin{itemize} 54\item \textbf{Probability distribution graph} - a series of points on a cartesian axis representing results of outcomes. $\Pr(X=x)$ is on $y$-axis, $x$ is on $x$ axis. 55\item \textbf{Mean $\mu$} or \textbf{expected value} \(E(X)\) - measure of central tendency. Also known as \textit{balance point}. Centre of a symmetrical distribution. 56\begin{align*} 57\overline{x} = \mu = E(X) &= \frac{\Sigma(xf)}{\Sigma(f)} \\ 58 &= \sum_{i=1}^n (x_i \cdot P(X=x_i)) \\ 59 &= \int_{-\infty}^{\infty} x\cdot f(x) \> dx \quad \text{(for pdf } f \text{)} 60 &= \sum_{-\infty}^{\infty} 61\end{align*} 62\item \textbf{Mode} - most popular value (has highest probability of \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution. 63\item \textbf{Median \(m\)} - the value of \(x\) such that \(\Pr(X \le m) = \Pr(X \ge m) = 0.5\). 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. 64 \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \] 65\item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. For distribution \(x_1 \mapsto p_1, x_2 \mapsto p_2, \dots, x_n \mapsto p_n\): 66\begin{align*} 67\sigma^2=\operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\ 68 &= \sum (x-\mu)^2\times \Pr(X=x) \\ 69 &= \sum x^2\times p(x) - \mu^2 70\end{align*} 71\item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$ 72\end{itemize} 73 74\subsubsection{Expectation theorems} 75 76\begin{align*} 77 E(aX \pm b) &= aE(X) \pm b \\ 78 E(z) &= z \\ 79 E(X+Y) &= E(X) + E(Y) \\ 80 E(X)^n &= \Sigma x^n \cdot p(x) \\ 81 &\ne[E(X)]^2 82\end{align*} 83 84 85\section{Binomial Theorem} 86 87\begin{align*} 88 (x+y)^n &= {n \choose0} x^n y^0 + {n \choose1} x^{n-1}y^1 + {n \choose2} x^{n-2}y^2 + \dots + {n \choose n-1}x^1 y^{n-1} + {n \choose n} x^0 y^n \\ 89 &= \sum_{k=0}^n {n \choose k} x^{n-k} y^k \\ 90 &= \sum_{k=0}^n {n \choose k} x^k y^{n-k} 91\end{align*} 92 93\begin{enumerate} 94\item powers of \(x\) decrease \(n \rightarrow0\) 95\item powers of \(y\) increase \(0\rightarrow n\) 96\item coefficients are given by \(n\)th row of Pascal's Triangle where \(n=0\) has one term 97\item Number of terms in \((x+a)^n\) expanded \& simplified is \(n+1\) 98\end{enumerate} 99 100 Combinations: \(^n\text{C}_r = {N\choose k}\) (binomial coefficient) 101\begin{itemize} 102\item Arrangements \({n \choose k} = \frac{n!}{(n-r)}\) 103\item Combinations \({n \choose k} = \frac{n!}{r!(n-r)!}\) 104\item Note \({n \choose k} = {n \choose k-1}\) 105\end{itemize} 106 107\subsubsection{Pascal's Triangle} 108 109\begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}} 110 n=\cr0&&&&&&&1&&&&&&\\ 1111&&&&&&1&&1&&&&&\\ 1122&&&&&1&&2&&1&&&&\\ 1133&&&&1&&3&&3&&1&&&\\ 1144&&&1&&4&&6&&4&&1&&\\ 1155&&1&&5&&10&&10&&5&&1&\\ 1166&1&&6&&15&&20&&15&&6&&1 117\end{tabular} 118 119\end{document}