+
+
+ \section{Statistics}
+
+ \subsection*{Probability}
+
+ \begin{align*}
+ \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
+ \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
+ \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
+ \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
+ \end{align*}
+
+ Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\
+
+ Independent events:
+ \begin{flalign*}
+ \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
+ \Pr(A|B) &= \Pr(A) \\
+ \Pr(B|A) &= \Pr(B)
+ \end{flalign*}
+
+ \subsection*{Combinatorics}
+
+ \begin{itemize}
+ \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
+ \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
+ \item Note \({n \choose k} = {n \choose k-1}\)
+ \end{itemize}
+
+ \subsection*{Distributions}
+
+ \subsubsection*{Mean \(\mu\)}
+
+ \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)
+
+ \begin{align*}
+ E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
+ &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
+ &= \int_\textbf{X} (x \cdot f(x)) \> dx
+ \end{align*}
+
+ \subsubsection*{Mode}
+
+ Most 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.
+
+ \subsubsection*{Median}
+
+ 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.
+
+ \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
+
+ \subsubsection*{Variance \(\sigma^2\)}
+
+ \begin{align*}
+ \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
+ &= \sum (x-\mu)^2 \times \Pr(X=x) \\
+ &= \sum x^2 \times p(x) - \mu^2 \\
+ &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
+ &= E\left[(X-\mu)^2\right]
+ \end{align*}
+
+ \subsubsection*{Standard deviation \(\sigma\)}
+
+ \begin{align*}
+ \sigma &= \operatorname{sd}(X) \\
+ &= \sqrt{\operatorname{Var}(X)}
+ \end{align*}
+
+ \subsection*{Binomial distributions}
+
+ Conditions for a \textit{binomial distribution}:
+ \begin{enumerate}
+ \item Two possible outcomes: \textbf{success} or \textbf{failure}
+ \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
+ \item Finite number \(n\) of independent trials
+ \end{enumerate}
+
+
+ \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}
+
+ \begin{align*}
+ \mu(X) &= np \\
+ \operatorname{Var}(X) &= np(1-p) \\
+ \sigma(X) &= \sqrt{np(1-p)} \\
+ \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
+ \end{align*}
+
+ \begin{cas}
+ Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input
+ \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
+ \item [x:] no. of successes
+ \item [numtrial:] no. of trials
+ \item [pos:] probability of success
+ \end{description}
+ \end{cas}
+
+ \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}
+
+ \begin{align*}
+ E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
+ \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
+ \end{align*}
+
+ \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]
+
+
+ \subsection*{Two random variables \(X, Y\)}
+
+ If \(X\) and \(Y\) are independent:
+ \begin{align*}
+ \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
+ \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
+ \end{align*}
+
+ \subsection*{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^{\frac{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*{Expectation theorems}
+
+ For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).
+
+ \begin{align*}
+ E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
+ E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
+ &\ne [E(X)]^n \\
+ E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
+ E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
+ E(X+Y) &= E(X) + E(Y) \tag{two variables}
+ \end{align*}
+
+ \subsection*{Sample mean}
+
+ Approximation of the \textbf{population mean} determined experimentally.
+
+ \[ \overline{x} = \dfrac{\Sigma x}{n} \]
+
+ where
+ \begin{description}[nosep, labelindent=0.5cm]
+ \item \(n\) is the size of the sample (number of sample points)
+ \item \(x\) is the value of a sample point
+ \end{description}
+
+ \begin{cas}
+ \begin{enumerate}[leftmargin=3mm]
+ \item Spreadsheet
+ \item In cell A1:\\ \path{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}
+ \end{cas}
+
+ \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 \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).
+
+ For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)
+
+ \begin{cas}
+
+ \begin{itemize}
+ \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
+ \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
+ \end{itemize}
+
+ \end{cas}
+
+ \subsection*{Normal distributions}
+
+
+ \[ Z = \frac{X - \mu}{\sigma} \]
+
+ Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
+ \(\text{mean} = \text{mode} = \text{median}\)
+
+ \begin{warning}
+ Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
+ \end{warning}
+
+ \pgfmathdeclarefunction{gauss}{2}{%
+ \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
+ }
+ \pgfkeys{/pgf/decoration/.cd,
+ distance/.initial=10pt
+ } \pgfdeclaredecoration{add dim}{final}{
+ \state{final}{%
+ \pgfmathsetmacro{\dist}{5pt*\pgfkeysvalueof{/pgf/decoration/distance}/abs(\pgfkeysvalueof{/pgf/decoration/distance})}
+ \pgfpathmoveto{\pgfpoint{0pt}{0pt}}
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+ \pgfpathmoveto{\pgfpoint{\pgfdecoratedpathlength}{0pt}}
+ \pgfpathlineto{\pgfpoint{(\pgfdecoratedpathlength}{2*\dist}}
+ \pgfsetarrowsstart{latex}
+ \pgfsetarrowsend{latex}
+ \pgfpathmoveto{\pgfpoint{0pt}{\dist}}
+ \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{\dist}}
+ \pgfusepath{stroke}
+ \pgfpathmoveto{\pgfpoint{0pt}{0pt}}
+ \pgfpathlineto{\pgfpoint{\pgfdecoratedpathlength}{0pt}}
+ }}
+ \tikzset{dim/.style args={#1,#2}{decoration={add dim,distance=#2},
+ decorate,
+ postaction={decorate,decoration={text along path,
+ raise=#2,
+ text align={align=center},
+ text={#1}}}}}
+ \begin{figure*}[hb]
+ \centering
+ \begin{tikzpicture}
+ \begin{axis}[every axis plot post/.style={
+ mark=none,domain=-3:3,samples=50,smooth},
+ axis x line=bottom,
+ axis y line=left,
+ enlargelimits=upper,
+ x=\textwidth/10,
+ ytick={0.55},
+ yticklabels={\(\frac{1}{\sigma \sqrt{2\pi}}\)},
+ xtick={-2,-1,0,1,2},
+ x tick label style = {font=\footnotesize},
+ xticklabels={\((\mu-2\sigma)\), \((\mu-\sigma)\), \(\mu\), \((\mu+\sigma)\), \((\mu+2\sigma)\)},
+ xlabel={\(x\)},
+ every axis x label/.style={at={(current axis.right of origin)},anchor=north west},
+ every axis y label/.style={at={(axis description cs:-0.02,0.2)}, anchor=south west, rotate=90},
+ ylabel={\(\Pr(X=x)\)}]
+ \addplot {gauss(0,0.75)};
+ \fill[red!30] (-3,0) -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-((x)^2)/(2*0.75^2))} -- (3,0) -- cycle;
+ \fill[darkgray!30] (3,0) -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (3,0) -- cycle;
+ \fill[lightgray!30] (-2,0) -- plot[id=f3,domain=-2:2,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (2,0) -- cycle;
+ \fill[white!30] (-1,0) -- plot[id=f3,domain=-1:1,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (1,0) -- cycle;
+ \begin{scope}[<->]
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+ \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.7\%};
+ \end{scope}
+ \begin{scope}[-, dashed, gray]
+ \draw (-1,0) -- (-1, 0.35);
+ \draw (1,0) -- (1, 0.35);
+ \draw (-2,0) -- (-2, 0.25);
+ \draw (2,0) -- (2, 0.25);
+ \draw (-3,0) -- (-3, 0.15);
+ \draw (3,0) -- (3, 0.15);
+ \end{scope}
+ \end{axis}
+ \begin{axis}[every axis plot post/.append style={
+ mark=none,domain=-3:3,samples=50,smooth},
+ axis x line=bottom,
+ enlargelimits=upper,
+ x=\textwidth/10,
+ xtick={-2,-1,0,1,2},
+ axis x line shift=30pt,
+ hide y axis,
+ x tick label style = {font=\footnotesize},
+ xlabel={\(Z\)},
+ every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
+ \addplot {gauss(0,0.75)};
+ \end{axis}
+ \end{tikzpicture}
+ \end{figure*}
+ \end{document}