From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 2 Sep 2019 13:46:59 +0900
Subject: Initial commit

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 macros/latex/contrib/statex2/statex2-example.tex | 232 +++++++++++++++++++++++
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+\documentclass[dvipsnames,usenames]{report}
+%\documentclass[dvipsnames,usenames,autobold]{report}
+\usepackage{statex2}
+\usepackage{shortvrb}
+\MakeShortVerb{@}
+% Examples
+\begin{document}
+
+Many accents have been re-defined
+
+@ c \c{c} \pi \cpi@ $$ c \c{c} \pi \cpi$$ %upright constants like the speed of light and 3.14159...
+
+@int \e{\im x} \d{x}@ $$\int \e{\im x} \d{x}$$ %\d{x}; also note new commands \e and \im
+
+@\^{\beta_1}=b_1@ $$\^{\beta_1}=b_1$$
+
+@\=x=\frac{1}{n}\sum x_i@ $$\=x=\frac{1}{n}\sum x_i$$  %also, \b{x}, but see \ol{x} below
+
+@\b{x} = \frac{1}{n} \wrap[()]{x_1 +\.+ x_n}@ $$\b{x} = \frac{1}{n} \wrap[()]{x_1 +\.+ x_n}$$
+
+Sometimes overline is better:  @\b{x} \vs \ol{x}@ $$\b{x} \vs \ol{x}$$
+
+And, underlines are nice too: @\ul{x}@ $$\ul{x}$$
+
+Derivatives and partial derivatives:
+
+@\deriv{x}{x^2+y^2}@ $$\deriv{x}{x^2+y^2}$$
+@\pderiv{x}{x^2+y^2}@ $$\pderiv{x}{x^2+y^2}$$
+
+Or, rather, in the order of @\frac@:
+
+@\derivf{x^2+y^2}{x}@ $$\derivf{x^2+y^2}{x}$$
+@\pderivf{x^2+y^2}{x}@ $$\pderivf{x^2+y^2}{x}$$
+
+A few other nice-to-haves:
+
+@\chisq@ $$\chisq$$
+
+@\Gamma[n+1]=n!@ $$\Gamma[n+1]=n!$$
+
+@\binom{n}{x}@ $$\binom{n}{x}$$ %provided by amsmath package
+
+@\e{x}@  $$\e{x}$$
+
+@\H_0: \mu=0 \vs \H_1: \mu \neq 0 (\neg \H_0) @ $$\H_0: \mu=0 \vs \H_1: \mu \neq 0 (\neg \H_0) $$ 
+
+@\logit \wrap{p} = \log \wrap{\frac{p}{1-p}}@ $$\logit \wrap{p} = \log \wrap{\frac{p}{1-p}}$$
+\pagebreak
+Common distributions along with other features follows:
+
+Normal Distribution
+
+@Z ~ \N{0}{1}, \where \E{Z}=0 \and \V{Z}=1@ $$Z ~ \N{0}{1}, \where \E{Z}=0 \and \V{Z}=1$$
+
+@\P{|Z|>z_\ha}=\alpha@ $$\P{|Z|>z_\ha}=\alpha$$
+
+@\pN[z]{0}{1}@ $$\pN[z]{0}{1}$$ 
+
+or, in general
+
+@\pN[z]{\mu}{\sd^2}@ $$\pN[z]{\mu}{\sd^2}$$
+
+Sometimes, we subscript the following operations:
+
+@\E[z]{Z}=0, \V[z]{Z}=1, \and \P[z]{|Z|>z_\ha}=\alpha@ 
+$$\E[z]{Z}=0, \V[z]{Z}=1, \and \P[z]{|Z|>z_\ha}=\alpha$$
+
+Multivariate Normal Distribution
+
+@\bm{X} ~ \N[p]{\bm{\mu}}{\sfsl{\Sigma}}@ $$\bm{X} ~ \N[p]{\bm{\mu}}{\sfsl{\Sigma}}$$ 
+%\bm provided by the bm package 
+
+Chi-square Distribution
+
+@Z_i \iid \N{0}{1}, \where i=1 ,\., n@ $$Z_i \iid \N{0}{1}, \where i=1 ,\., n$$
+
+@\chisq = \sum_i Z_i^2 ~ \Chi{n}@ $$\chisq = \sum_i Z_i^2 ~ \Chi{n}$$
+
+@\pChi[z]{n}@ $$\pChi[z]{n}$$
+
+t Distribution
+
+@\frac{\N{0}{1}}{\sqrt{\frac{\Chisq{n}}{n}}} ~ \t{n}@ 
+$$\frac{\N{0}{1}}{\sqrt{\frac{\Chisq{n}}{n}}} ~ \t{n}$$
+\pagebreak
+F Distribution
+    
+@X_i, Y_{\~i} \iid \N{0}{1} \where i=1 ,\., n; \~i=1 ,\., m \and \V{X_i, Y_{\~i}}=\sd_{xy}=0@ 
+$$X_i, Y_{\~i} \iid \N{0}{1} \where i=1 ,\., n; \~i=1 ,\., m \and \V{X_i, Y_{\~i}}=\sd_{xy}=0$$
+
+@\chisq_x = \sum_i X_i^2 ~ \Chi{n}@ $$\chisq_x = \sum_i X_i^2 ~ \Chi{n}$$
+
+@\chisq_y = \sum_{\~i} Y_{\~i}^2 ~ \Chi{m}@ $$\chisq_y = \sum_{\~i} Y_{\~i}^2 ~ \Chi{m}$$
+
+@\frac{\chisq_x}{\chisq_y} ~ \F{n}{m}@ $$\frac{\chisq_x}{\chisq_y} ~ \F{n}{m}$$
+
+Beta Distribution
+
+@B=\frac{\frac{n}{m}F}{1+\frac{n}{m}F} ~ \Bet{\frac{n}{2}}{\frac{m}{2}}@ 
+$$B=\frac{\frac{n}{m}F}{1+\frac{n}{m}F} ~ \Bet{\frac{n}{2}}{\frac{m}{2}}$$
+
+@\pBet{\alpha}{\beta}@ $$\pBet{\alpha}{\beta}$$ 
+
+Gamma Distribution
+
+@G ~ \Gam{\alpha}{\beta}@ $$G ~ \Gam{\alpha}{\beta}$$
+
+@\pGam{\alpha}{\beta}@ $$\pGam{\alpha}{\beta}$$
+
+Cauchy Distribution
+
+@C ~ \Cau{\theta}{\nu}@ $$C ~ \Cau{\theta}{\nu}$$
+
+@\pCau{\theta}{\nu}@ $$\pCau{\theta}{\nu}$$
+
+Uniform Distribution
+
+@X ~ \U{0, 1}@ $$X ~ \U{0, 1}$$
+
+@\pU{0}{1}@ $$\pU{0}{1}$$
+
+or, in general
+
+@\pU{a}{b}@ $$\pU{a}{b}$$
+
+Exponential Distribution
+
+@X ~ \Exp{\lambda}@ $$X ~ \Exp{\lambda}$$
+
+@\pExp{\lambda}@ $$\pExp{\lambda}$$
+
+Hotelling's $T^2$ Distribution
+
+@X ~ \Tsq{\nu_1}{\nu_2}@ $$X ~ \Tsq{\nu_1}{\nu_2}$$
+
+Inverse Chi-square Distribution
+
+@X ~ \IC{\nu}@ $$X ~ \IC{\nu}$$
+
+Inverse Gamma Distribution
+
+@X ~ \IG{\alpha}{\beta}@ $$X ~ \IG{\alpha}{\beta}$$
+
+Pareto Distribution
+
+@X ~ \Par{\alpha}{\beta}@ $$X ~ \Par{\alpha}{\beta}$$
+
+@\pPar{\alpha}{\beta}@ $$\pPar{\alpha}{\beta}$$
+
+Wishart Distribution
+
+@\sfsl{X} ~ \W{\nu}{\sfsl{S}}@ $$\sfsl{X} ~ \W{\nu}{\sfsl{S}}$$
+
+Inverse Wishart Distribution
+
+@\sfsl{X} ~ \IW{\nu}{\sfsl{S^{-1}}}@ $$\sfsl{X} ~ \IW{\nu}{\sfsl{S^{-1}}}$$
+
+Binomial Distribution
+
+@X ~ \Bin{n}{p}@ $$X ~ \Bin{n}{p}$$
+
+%@\pBin{n}{p}@ $$\pBin{n}{p}$$
+
+Bernoulli Distribution
+
+@X ~ \B{p}@ $$X ~ \B{p}$$
+
+Beta-Binomial Distribution
+
+@X ~ \BB{p}@ $$X ~ \BB{p}$$
+
+%@\pBB{n}{\alpha}{\beta}@ $$\pBB{n}{\alpha}{\beta}$$
+
+Negative-Binomial Distribution
+
+@X ~ \NB{n}{p}@ $$X ~ \NB{n}{p}$$
+
+Hypergeometric Distribution
+
+@X ~ \HG{n}{M}{N}@ $$X ~ \HG{n}{M}{N}$$
+
+Poisson Distribution
+
+@X ~ \Poi{\mu}@ $$X ~ \Poi{\mu}$$
+
+%@\pPoi{\mu}@ $$\pPoi{\mu}$$
+
+Dirichlet Distribution
+
+@\bm{X} ~ \Dir{\alpha_1 \. \alpha_k}@ $$\bm{X} ~ \Dir{\alpha_1 \. \alpha_k}$$
+
+Multinomial Distribution
+
+@\bm{X} ~ \M{n}{\alpha_1 \. \alpha_k}@ $$\bm{X} ~ \M{n}{\alpha_1 \. \alpha_k}$$
+
+\pagebreak
+
+To compute critical values for the Normal distribution, create the
+NCRIT program for your TI-83 (or equivalent) calculator.  At each step, the 
+calculator display is shown, followed by what you should do (\Rect\ is the 
+cursor):\\
+\Rect\\
+\Prgm\to@NEW@\to@1:Create New@\\
+@Name=@\Rect\\
+NCRIT\Enter\\
+@:@\Rect\\
+\Prgm\to@I/O@\to@2:Prompt@\\
+@:Prompt@ \Rect\\
+\Alpha[A],\Alpha[T]\Enter\\
+@:@\Rect\\
+\Distr\to@DISTR@\to@3:invNorm(@\\
+@:invNorm(@\Rect\\
+1-(\Alpha[A]$\div$\Alpha[T]))\Sto\Alpha[C]\Enter\\
+@:@\Rect\\
+\Prgm\to@I/O@\to@3:Disp@\\
+@:Disp@ \Rect\\
+\Alpha[C]\Enter\\
+@:@\Rect\\
+\Quit\\
+
+Suppose @A@ is $\alpha$ and @T@ is the number of tails.  To run the program:\\
+\Rect\\
+\Prgm\to@EXEC@\to@NCRIT@\\
+@prgmNCRIT@\Rect\\
+\Enter\\
+@A=?@\Rect\\
+0.05\Enter\\
+@T=?@\Rect\\
+2\Enter\\
+@1.959963986@
+\end{document}
+
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