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+\usepackage{array}
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\usepackage{pgfplots}
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+\usepackage{keystroke}
+\usepackage{listings}
+\usepackage{xcolor} % used only to show the phantomed stuff
+\definecolor{cas}{HTML}{e6f0fe}
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\fancyhead[LO,LE]{Unit 3 Methods Statistics}
\begin{itemize}
\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.
- \item \textbf{Mean $\mu$} - measure of central tendency. \textit{Balance point} or \textit{expected value} of a distribution. Centre of a symmetrical distribution.
- \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. Represented by $\sigma^2=\operatorname{Var}(x) = \sum (x=\mu)^2 \times p(x) = \sum (x-\mu)^2 \times \Pr(X=x)$. Alternatively: $\sigma^2 = \operatorname{Var}(X) = \sum x^2 \times p(x) - \mu^2$
+ \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.
+ \begin{align*}
+ \overline{x} = \mu = E(X) &= \frac{\Sigma(xf)}{\Sigma(f)} \\
+ &= \sum_{i=1}^n (x_i \cdot P(X=x_i)) \\
+ &= \int_{-\infty}^{\infty} x\cdot f(x) \> dx \quad \text{(for pdf } f \text{)}
+ &= \sum_{-\infty}^{\infty}
+ \end{align*}
+ \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.
+ \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.
+ \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]
+ \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. For distribution \(x_1 \mapsto p_1, x_2 \mapsto p_2, \dots, x_n \mapsto p_n\):
+ \begin{align*}
+ \sigma^2=\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
+ \end{align*}
\item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$
\end{itemize}
+ \subsubsection{Expectation theorems}
+
+ \begin{align*}
+ E(aX \pm b) &= aE(X) \pm b \\
+ E(z) &= z \\
+ E(X+Y) &= E(X) + E(Y) \\
+ E(X)^n &= \Sigma x^n \cdot p(x) \\
+ &\ne [E(X)]^2
+ \end{align*}
+
+
+ \section{Binomial Theorem}
+
+ \begin{align*}
+ (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 \\
+ &= \sum_{k=0}^n {n \choose k} x^{n-k} y^k \\
+ &= \sum_{k=0}^n {n \choose k} x^k y^{n-k}
+ \end{align*}
+
+ \begin{enumerate}
+ \item powers of \(x\) decrease \(n \rightarrow 0\)
+ \item powers of \(y\) increase \(0 \rightarrow n\)
+ \item coefficients are given by \(n\)th row of Pascal's Triangle where \(n=0\) has one term
+ \item Number of terms in \((x+a)^n\) expanded \& simplified is \(n+1\)
+ \end{enumerate}
+
+ Combinations: \(^n\text{C}_r = {N\choose k}\) (binomial coefficient)
+ \begin{itemize}
+ \item Arrangements \({n \choose k} = \frac{n!}{(n-r)}\)
+ \item Combinations \({n \choose k} = \frac{n!}{r!(n-r)!}\)
+ \item Note \({n \choose k} = {n \choose k-1}\)
+ \end{itemize}
+
+ \subsubsection{Pascal's Triangle}
+
+ \begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}}
+ n=\cr0&&&&&&&1&&&&&&\\
+ 1&&&&&&1&&1&&&&&\\
+ 2&&&&&1&&2&&1&&&&\\
+ 3&&&&1&&3&&3&&1&&&\\
+ 4&&&1&&4&&6&&4&&1&&\\
+ 5&&1&&5&&10&&10&&5&&1&\\
+ 6&1&&6&&15&&20&&15&&6&&1
+ \end{tabular}
+
+ \colorbox{cas}{On CAS:} (soft keys) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr);
+
+ \section{Binomial distributions}
+
+ (aka Bernoulli distributions)
+
+ \begin{align*}
+ \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\
+ &= {n \choose x} p^x q^{n-x}
+ \end{align*}
+
+ \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}
+
+ If these conditions are met, then it is a Binomial Random Variable. This variable is said to have a \textit{binomial probability distribution}.
+
+ \begin{itemize}
+ \item \(n\) is the number of trials
+ \item There are two possible outcomes: \(S\) or \(F\)
+ \item \(\Pr(\text{success}) = p\)
+ \item \(\Pr(\text{failure}) = 1-p = q\)
+ \item Shorthand notation: \(X \sim \operatorname{Bi}(n,p)\)
+ \end{itemize}
+
+ \colorbox{cas}{On CAS:} Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf; \\
+ Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success)
+
+ \subsection{Applications of binomial distributions}
+
+ \[ \Pr(X \ge a) = 1 - \Pr(X < a) \]
+
+ \subsection{Expected value of a binomial distribution}
+
+ \[ E(X \sim \operatorname{Bi}(n,p))=np \]
+
+ \subsection{Variance}
+
+ \[ \sigma^2(X) = np(1-p) \]
+
+ \subsection{Standard deviation}
+
+ \[ \sigma(X) = \sqrt{np(1-p)} \]
+
\end{document}