From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 21 Aug 2019 05:02:44 +0000 (+1000)
Subject: [spec] add statistics notes up to 15E
X-Git-Tag: yr12~60
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/a11aed77bd96142b075834cad47f46d795a92f5e

[spec] add statistics notes up to 15E
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+\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}