1\documentclass[methods-collated.tex]{subfiles} 2 3\begin{document} 4 5\section{Statistics} 6 7\subsection*{Probability} 8 9\begin{align*} 10\Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\ 11\Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\ 12\Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\ 13\Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) 14\end{align*} 15 16Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\ 17 18Independent events: 19\begin{flalign*} 20\quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\ 21\Pr(A|B) &= \Pr(A) \\ 22\Pr(B|A) &= \Pr(B) 23\end{flalign*} 24 25\subsection*{Combinatorics} 26 27\begin{itemize} 28\item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\) 29\item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\) 30\item Note \({n \choose k} = {n \choose k-1}\) 31\end{itemize} 32 33\subsection*{Distributions} 34 35\subsubsection*{Mean \(\mu\)} 36 37\textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\) 38 39\begin{align*} 40 E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f}\tag{\(f =\) absolute frequency} \\ 41 &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right]\tag{discrete}\\ 42 &= \int_\textbf{X} (x \cdot f(x)) \> dx 43\end{align*} 44 45\subsubsection*{Mode} 46 47Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution. 48 49\subsubsection*{Median} 50 51If \(m > 0.5\), 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. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5. 52 53\[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \] 54 55\subsubsection*{Variance \(\sigma^2\)} 56 57\begin{align*} 58\operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\ 59 &= \sum (x-\mu)^2\times \Pr(X=x) \\ 60 &= \sum x^2\times p(x) - \mu^2 \\ 61 &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2 \\ 62 &= E\left[(X-\mu)^2\right] 63\end{align*} 64 65\subsubsection*{Standard deviation \(\sigma\)} 66 67\begin{align*} 68\sigma &= \operatorname{sd}(X) \\ 69 &= \sqrt{\operatorname{Var}(X)} 70\end{align*} 71 72\subsection*{Binomial distributions} 73 74Conditions for a \textit{binomial distribution}: 75\begin{enumerate} 76\item Two possible outcomes: \textbf{success} or \textbf{failure} 77\item \(\Pr(\text{success})\) (=\(p\)) is constant across trials 78\item Finite number \(n\) of independent trials 79\end{enumerate} 80 81 82\subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)} 83 84\begin{align*} 85\mu(X) &= np \\ 86\operatorname{Var}(X) &= np(1-p) \\ 87\sigma(X) &= \sqrt{np(1-p)} \\ 88\Pr(X=x) &= {n \choose x}\cdot p^x \cdot (1-p)^{n-x} 89\end{align*} 90 91\begin{cas} 92 Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; 93\begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont] 94\item[x:] no. of successes 95\item[numtrial:] no. of trials 96\item[pos:] probability of success 97\end{description} 98\end{cas} 99 100\subsection*{Continuous random variables} 101 102A continuous random variable \(X\) has a pdf \(f\) such that: 103 104\begin{enumerate} 105\item \(f(x) \ge0\forall x \) 106\item \(\int^\infty_{-\infty} f(x) \> dx = 1\) 107\end{enumerate} 108 109\begin{align*} 110 E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\ 111\operatorname{Var}(X) &= E\left[(X-\mu)^2\right] 112\end{align*} 113 114\[\Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \] 115 116\begin{cas} 117 Define piecewise functions: \\ 118 Math3 \(\rightarrow\) 119% TODO: finish this section 120\end{cas} 121 122\subsection*{Two random variables \(X, Y\)} 123 124If \(X\) and \(Y\) are independent: 125\begin{align*} 126\operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\ 127\operatorname{Var}(aX \pm bY \pm c) &= a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) 128\end{align*} 129 130\subsection*{Linear functions \(X \rightarrow aX+b\)} 131 132\begin{align*} 133\Pr(Y \le y) &= \Pr(aX+b \le y) \\ 134 &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\ 135 &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx 136\end{align*} 137 138\begin{align*} 139\textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\ 140\textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2\operatorname{Var}(X) \\ 141\end{align*} 142 143\subsection*{Expectation theorems} 144 145For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\). 146 147\begin{align*} 148 E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\ 149 E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\ 150 &\ne[E(X)]^n \\ 151 E(aX \pm b) &= aE(X) \pm b \tag{linear} \\ 152 E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\ 153 E(X+Y) &= E(X) + E(Y) \tag{two variables} 154\end{align*} 155 156\begin{figure*}[hb] 157\centering 158\include{../spec/normal-dist-graph} 159\end{figure*} 160 161\subsection*{Sample mean} 162 163Approximation of the \textbf{population mean} determined experimentally. 164 165\[\overline{x} = \dfrac{\Sigma x}{n} \] 166 167where 168\begin{description}[nosep, labelindent=0.5cm] 169\item \(n\) is the size of the sample (number of sample points) 170\item \(x\) is the value of a sample point 171\end{description} 172 173\begin{cas} 174\begin{enumerate}[leftmargin=3mm] 175\item Spreadsheet 176\item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))} 177\item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range 178\item Input range as A1:An where \(n\) is the number of samples 179\item Graph \(\rightarrow\) Histogram 180\end{enumerate} 181\end{cas} 182 183\subsubsection*{Sample size of \(n\)} 184 185\[\overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \] 186 187Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)). 188 189For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\) 190 191\begin{cas} 192 193\begin{itemize} 194\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 195\item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable 196\end{itemize} 197 198\end{cas} 199 200\subsection*{Normal distributions} 201 202 203\[ Z = \frac{X - \mu}{\sigma} \] 204 205Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\ 206\(\text{mean} = \text{mode} = \text{median}\) 207 208\begin{warning} 209 Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair. 210\end{warning} 211 212\subsection*{Confidence intervals} 213 214\begin{itemize} 215\item \textbf{Point estimate:} single-valued estimate of the population mean from the value of the sample mean \(\overline{x}\) 216\item \textbf{Interval estimate:} confidence interval for population mean \(\mu\) 217\item \(C\)\% confidence interval \(\implies\) \(C\)\% of samples will contain population mean \(\mu\) 218\end{itemize} 219 220\subsubsection*{95\% confidence interval} 221 222For 95\% c.i. of population mean \(\mu\): 223 224\[ x \in \left(\overline{x}\pm1.96\dfrac{\sigma}{\sqrt{n}}\right)\] 225 226where: 227\begin{description}[nosep, labelindent=0.5cm] 228\item \(\overline{x}\) is the sample mean 229\item \(\sigma\) is the population sd 230\item \(n\) is the sample size from which \(\overline{x}\) was calculated 231\end{description} 232 233\begin{cas} 234 Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\ 235 Set \textit{Type = One-Sample Z Int} \\ \-\hspace{1em} and select \textit{Variable} 236\end{cas} 237 238\subsection*{Margin of error} 239 240For 95\% confidence interval of \(\mu\): 241\begin{align*} 242 M &= 1.96\times \dfrac{\sigma}{\sqrt{n}} \\ 243 &= \dfrac{1}{2}\times \text{width of c.i.} \\ 244\implies n &= \left( \dfrac{1.96\sigma}{M}\right)^2 245\end{align*} 246 247Always round \(n\) up to a whole number of samples. 248 249\subsection*{General case} 250 251For \(C\)\% c.i. of population mean \(\mu\): 252 253\[ x \in \left( \overline{x}\pm k \dfrac{\sigma}{\sqrt{n}}\right) \] 254\hfill where \(k\) is such that \(\Pr(-k < Z < k) = \frac{C}{100}\) 255 256\begin{cas} 257 Menu \(\rightarrow\) Stats \(\rightarrow\) Calc \(\rightarrow\) Interval \\ 258 Set \textit{Type = One-\colorbox{important}{Prop} Z Int} \\ 259 Input x \(= \hat{p} * n\) 260\end{cas} 261 262\subsection*{Confidence interval for multiple trials} 263 264For a set of \(n\) confidence intervals (samples), there is \(0.95^n\) chance that all \(n\) intervals contain the population mean \(\mu\). 265 266\end{document}