methods / statistics.texon commit [chem] start organic reactions summary document (5447e6c)
   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\usepackage{keystroke}
  11\usepackage{listings}
  12\usepackage{xcolor} % used only to show the phantomed stuff
  13\definecolor{cas}{HTML}{e6f0fe}
  14\usepackage{mathtools}
  15
  16\pagestyle{fancy}
  17\fancyhead[LO,LE]{Unit 3 Methods --- Statistics}
  18\fancyhead[CO,CE]{Andrew Lorimer}
  19
  20\setlength\parindent{0pt}
  21
  22\begin{document}
  23
  24  \title{Statistics}
  25  \author{}
  26  \date{}
  27  %\maketitle
  28
  29  \section{Probability}
  30  
  31  \subsection*{Probability theorems}
  32
  33  \begin{align*}
  34    \textbf{Union:} &&\Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
  35    \textbf{Multiplication theorem:} &&\Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
  36    \textbf{Conditional:} &&\Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
  37    \textbf{Law of total probability:} &&\Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) \\
  38  \end{align*}
  39
  40  Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
  41  
  42  Independent events:
  43  \begin{flalign*}
  44    \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
  45    \Pr(A|B) &= \Pr(A) \\
  46    \Pr(B|A) &= \Pr(B)
  47  \end{flalign*}
  48
  49  \subsection*{Discrete random distributions}
  50
  51  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}.
  52
  53  \subsubsection*{Discrete probability distributions}
  54  
  55  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.
  56
  57  \begin{itemize}
  58    \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.
  59    \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.
  60      \begin{align*}
  61        \overline{x} = \mu = E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{where \(f =\) absolute frequency} \\
  62        &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{for \(n\) values of \(x\)}\\
  63        &= \int_{-\infty}^{\infty} (x\cdot f(x)) \> dx \tag{for pdf \(f\)}
  64      \end{align*}
  65    \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.
  66    \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. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.
  67      \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
  68    \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. For distribution \(x_1 \mapsto p_1, x_2 \mapsto p_2, \dots, x_n \mapsto p_n\):
  69      \begin{align*}
  70        \sigma^2=\operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
  71        &= \sum (x-\mu)^2 \times \Pr(X=x) \\
  72        &= \sum x^2 \times p(x) - \mu^2 \\
  73        &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
  74      \end{align*}
  75    \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance:
  76      \begin{align*}
  77        \sigma &= \operatorname{sd}(X) \\
  78        &= \sqrt{\operatorname{Var}(X)}
  79      \end{align*}
  80  \end{itemize}
  81
  82  \subsubsection*{Expectation theorems}
  83
  84  For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
  85
  86  \begin{align*}
  87    E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear function} \\
  88    &\ne [E(X)]^n \\
  89    E(aX \pm b) &= aE(X) \pm b \tag{linear function} \\
  90    E(b) &= b \tag{for constant \(b \in \mathbb{R}\)}\\
  91    E(X+Y) &= E(X) + E(Y) \tag{for two random variables}
  92  \end{align*}
  93
  94  \subsubsection*{Variance theorems}
  95
  96  \[ \operatorname{Var}(aX \pm bY \pm c) = a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \]
  97
  98  \section{Binomial Theorem}
  99
 100  \begin{align*}
 101    (x+y)^n &= {n \choose 0} x^n y^0 + {n \choose 1} x^{n-1}y^1 + {n \choose 2} x^{n-2}y^2 + \dots + {n \choose n-1}x^1 y^{n-1} + {n \choose n} x^0 y^n \\
 102    &= \sum_{k=0}^n {n \choose k} x^{n-k} y^k \\
 103    &= \sum_{k=0}^n {n \choose k} x^k y^{n-k}
 104  \end{align*}
 105
 106  \subsubsection*{Patterns}
 107  \begin{enumerate}
 108    \item powers of \(x\) decrease \(n \rightarrow 0\)
 109    \item powers of \(y\) increase \(0 \rightarrow n\)
 110    \item coefficients are given by \(n\)th row of Pascal's Triangle where \(n=0\) has one term
 111    \item Number of terms in \((x+a)^n\) expanded \& simplified is \(n+1\)
 112  \end{enumerate}
 113
 114  \subsubsection*{Combinatorics}
 115
 116  \[ \text{Binomial coefficient:} \quad ^n\text{C}_r = {N\choose k} \]
 117
 118  \begin{itemize}
 119    \item Arrangements \({n \choose k} = \frac{n!}{(n-r)}\)
 120    \item Combinations \({n \choose k} = \frac{n!}{r!(n-r)!}\)
 121    \item Note \({n \choose k} = {n \choose k-1}\)
 122  \end{itemize}
 123
 124  \colorbox{cas}{On CAS:} (soft keyboard) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr);
 125
 126  \subsubsection*{Pascal's Triangle}
 127
 128  \begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}}
 129    n=\cr0&&&&&&&1&&&&&&\\
 130    1&&&&&&1&&1&&&&&\\
 131    2&&&&&1&&2&&1&&&&\\
 132    3&&&&1&&3&&3&&1&&&\\
 133    4&&&1&&4&&6&&4&&1&&\\
 134    5&&1&&5&&10&&10&&5&&1&\\
 135    6&1&&6&&15&&20&&15&&6&&1
 136  \end{tabular}
 137
 138  \section{Binomial distributions}
 139
 140  (aka Bernoulli distributions)
 141
 142  \begin{align*}
 143    \text{Defined by} \quad X &\sim \operatorname{Bi}(n,p) \\
 144    \implies \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\
 145    &= {n \choose x} p^x q^{n-x}
 146  \end{align*}
 147
 148  where:
 149  \begin{description}
 150    \item \(n\) is the number of trials
 151    \item There are two possible outcomes: \(S\) or \(F\)
 152    \item \(\Pr(\text{success}) = p\)
 153    \item \(\Pr(\text{failure}) = 1-p = q\)
 154  \end{description}
 155   
 156  \subsection*{Conditions for a binomial variable/distribution}
 157  \begin{enumerate}
 158    \item Two possible outcomes: \textbf{success} or \textbf{failure}
 159    \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
 160    \item Finite number \(n\) of independent trials
 161  \end{enumerate}
 162
 163  \subsection*{\colorbox{cas}{Solve on CAS}}
 164  
 165  Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf;
 166
 167  \hspace{2em} Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success)
 168
 169  \subsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
 170
 171  \begin{align*}
 172    \textbf{Mean} \hspace{-4cm} &&\mu(X) &= np \\
 173    \textbf{Variance} \hspace{-4cm} &&\sigma^2(X) &= np(1-p) \\
 174    \textbf{s.d.} \hspace{-4cm} &&\sigma(X) &= \sqrt{np(1-p)}
 175  \end{align*}
 176
 177  \subsection*{Applications of binomial distributions}
 178
 179  \[ \Pr(X \ge a) = 1 - \Pr(X < a) \]
 180
 181  \section{Continuous probability}
 182
 183  \subsection*{Continuous random variables}
 184
 185  \begin{itemize}
 186    \item a variable that can take any real value in an interval
 187  \end{itemize}
 188
 189  \subsection*{Probability density functions}
 190
 191  \begin{itemize}
 192    \item area under curve \( = 1 \implies \int f(x) \> dx = 1\)
 193    \item \(f(x) \ge 0 \forall x\)
 194    \item pdfs may be linear
 195    \item must show sections where \(f(x) = 0\) (use open/closed circles)
 196  \end{itemize}
 197
 198  \[ Pr(a \le X \le b) = \int^b_a f(x) \> dx \]
 199
 200  \colorbox{cas}{On CAS:} Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;normCdf;.
 201
 202  For function in domain \(a \le x \le b\):
 203
 204  \[ \operatorname{E}(X) = \int^b_a x f(x) \> dx \]
 205
 206  \[ \operatorname{sd}(X) = \sqrt{\operatorname{Var}(X)} = \sqrt{\operatorname{E}(X^2)-[\operatorname{E}(X)]^2} \]
 207
 208\end{document}