1\documentclass[a4paper]{article} 2\usepackage[a4paper,margin=2cm]{geometry} 3\usepackage{amsmath} 4\usepackage{amssymb} 5\usepackage{tcolorbox} 6\usepackage{fancyhdr} 7\usepackage{pgfplots} 8\usepackage{tabularx} 9 10\pagestyle{fancy} 11\fancyhead[LO,LE]{Unit 3 Methods Statistics} 12\fancyhead[CO,CE]{Andrew Lorimer} 13 14\setlength\parindent{0pt} 15 16\begin{document} 17 18 \title{Statistics} 19 \author{} 20 \date{} 21 \maketitle 22 23 \section{Probability} 24 25 \[ \Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B) \] 26 \[ \Pr(A \cup B) = 0 \tag{mutually exclusive} \] 27 28 \section{Conditional probability} 29 30 \[ \Pr(A|B) = \frac{\Pr(A \cap B)}{\Pr(B)} \quad \text{where } \Pr(B) \ne 0 \] 31 32 \[ \Pr(A) = \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) \tag{law of total probability} \] 33 34 \[ \Pr(A \cap B) = \Pr(A|B) \times \Pr(B) \tag{multiplication theorem} \] 35 36 For independent events: 37 38 \begin{itemize} 39 \item \(\Pr(A \cap B) = \Pr(A) \times \Pr(B)\) 40 \item \(\Pr(A|B) = \Pr(A)\) 41 \item \(\Pr(B|A) = \Pr(B)\) 42 \end{itemize} 43 44 \subsection{Discrete random distributions} 45 46 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}. 47 48 \subsubsection{Discrete probability distributions} 49 50 If an activity has outcomes whose probability values are all positive and less than one ($\implies 0 \le p(x) \le 1$), 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. 51 52 \begin{itemize} 53 \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. 54 \item \textbf{Mean $\mu$} - measure of central tendency. \textit{Balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution. 55 \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. Represented by $\sigma^2=\operatorname{Var}(x) = \sum (x=\mu)^2 \times p(x) = \sum (x-\mu)^2 \times \Pr(X=x)$. Alternatively: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$ 56 \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$ 57 \end{itemize} 58 59\end{document}