-\documentclass[a4paper, twocolumn]{article}
+\documentclass[a4paper]{article}
\usepackage[dvipsnames, table]{xcolor}
\usepackage{adjustbox}
\usepackage{amsmath}
\usepackage{pgfplots}
\usepackage{pst-plot}
\usepackage{standalone}
+\usepackage{subfiles}
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\date{}
\maketitle
+\begin{multicols}{2}
+
\section{Functions}
\input{circ-functions}
\input{calculus}
+ \subfile{statistics-ref}
+ \end{multicols}
- \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}
-
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- \end{document}
+\end{document}