+\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}
+
+\pagestyle{fancy}
+\fancyhead[LO,LE]{Unit 4 Specialist --- Statistics}
+\fancyhead[CO,CE]{Andrew Lorimer}
+
+\setlength\parindent{0pt}
+
+\begin{document}
+
+ \title{Statistics}
+ \author{}
+ \date{}
+ \maketitle
+
+ \section{Linear combinations of random variables}
+
+ \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}
+
+ \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
+
+ \subsubsection*{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^{\dfrac{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*{Linear combination of two random variables}
+
+ \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}\\
+ \end{align*}
+
+ \section{Sample mean}
+
+ \[ \overline{x} = \dfrac{\Sigma x}{n} \]
+
+ where \(n\) is the size of the sample (number of sample points)
+
+ \subsubsection*{\colorbox{cas}{On CAS:}}
+
+ \begin{enumerate}
+ \item Spreadsheet
+ \item In cell A1: \verb;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}
+
+ \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 \(\frac{\sigma}{\sqrt{n}}\)
+
+
+
+\end{document}