LaTeX-examples/documents/normal-distribution-z/normal-distribution-z.tex

\documentclass[a4paper]{article}
\usepackage[margin=20mm]{geometry}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}
\usepackage{amssymb,amsmath}
\usepackage{slashbox}
\usepackage{booktabs}
\usepackage[table,x11names]{xcolor}
\usepackage{pgfplots}

\setlength{\intextsep}{1pt}

\begin{document}
    \pagenumbering{gobble}
    \section*{Percent point function (PPF) of the normal distribution}

\pgfmathdeclarefunction{gauss}{3}{%
  \pgfmathparse{1/(#3*sqrt(2*pi))*exp(-((#1-#2)^2)/(2*#3^2))}%
}

\begin{figure}[!h]
\centering
\begin{tikzpicture}
\begin{axis}[
  no markers,
  domain=0:6,
  samples=100,
  ymin=0,
  axis lines*=left,
  xlabel=$x$,
  every axis y label/.style={at=(current axis.above origin),anchor=south},
  every axis x label/.style={at=(current axis.right of origin),anchor=west},
  height=4cm,
  width=12cm,
  xtick=\empty,
  ytick=\empty,
  enlargelimits=false,
  clip=false,
  axis on top,
  grid = major,
  hide y axis
  ]

 \addplot [very thick,cyan!50!black] {gauss(x, 3, 1)};

\pgfmathsetmacro\valueA{gauss(1,3,1)}
\pgfmathsetmacro\valueB{gauss(2,3,1)}
\draw [gray] (axis cs:1,0) -- (axis cs:1,\valueA)
    (axis cs:5,0) -- (axis cs:5,\valueA);
\draw [gray] (axis cs:2,0) -- (axis cs:2,\valueB)
    (axis cs:4,0) -- (axis cs:4,\valueB);
\draw [yshift=1.4cm, latex-latex](axis cs:2, 0) -- node [fill=white] {$0.683$} (axis cs:4, 0);
\draw [yshift=0.3cm, latex-latex](axis cs:1, 0) -- node [fill=white] {$0.954$} (axis cs:5, 0);

\node[below] at (axis cs:1, 0)  {$\mu - 2\sigma$};
\node[below] at (axis cs:2, 0)  {$\mu - \sigma$};
\node[below] at (axis cs:3, 0)  {$\mu$};
\end{axis}
\end{tikzpicture}
\end{figure}

\begin{table}[!h]
    \centering
    \begin{tabular}{c|ccccc|ccccc}
    \toprule
    \backslashbox{$x$}{$\Delta x$}  & \textbf{0.00} & \textbf{0.01} & \textbf{0.02} & \textbf{0.03} & \textbf{0.04} & \textbf{0.05} & \textbf{0.06} & \textbf{0.07} & \textbf{0.08} & \textbf{0.09} \\\midrule
\textbf{0.0} & -inf & -2.3263 & -2.0537 & -1.8808 & -1.7507 & -1.6449 & -1.5548 & -1.4758 & -1.4051 & -1.3408\\
\textbf{0.1} & -1.2816 & -1.2265 & -1.1750 & -1.1264 & -1.0803 & -1.0364 & -0.9945 & -0.9542 & -0.9154 & -0.8779\\
\textbf{0.2} & -0.8416 & -0.8064 & -0.7722 & -0.7388 & -0.7063 & -0.6745 & -0.6433 & -0.6128 & -0.5828 & -0.5534\\
\textbf{0.3} & -0.5244 & -0.4959 & -0.4677 & -0.4399 & -0.4125 & -0.3853 & -0.3585 & -0.3319 & -0.3055 & -0.2793\\
\textbf{0.4} & -0.2533 & -0.2275 & -0.2019 & -0.1764 & -0.1510 & -0.1257 & -0.1004 & -0.0753 & -0.0502 & -0.0251\\
\textbf{0.5} & 0.0000 & 0.0251 & 0.0502 & 0.0753 & 0.1004 & 0.1257 & 0.1510 & 0.1764 & 0.2019 & 0.2275\\
\textbf{0.6} & 0.2533 & 0.2793 & 0.3055 & 0.3319 & 0.3585 & 0.3853 & 0.4125 & 0.4399 & 0.4677 & 0.4959\\
\textbf{0.7} & 0.5244 & 0.5534 & 0.5828 & 0.6128 & 0.6433 & 0.6745 & 0.7063 & 0.7388 & 0.7722 & 0.8064\\
\textbf{0.8} & 0.8416 & 0.8779 & 0.9154 & 0.9542 & 0.9945 & 1.0364 & 1.0803 & 1.1264 & 1.1750 & 1.2265\\
\textbf{0.9} & 1.2816 & 1.3408 & 1.4051 & 1.4758 & 1.5548 & 1.6449 & 1.7507 & 1.8808 & 2.0537 & 2.3263\\
        \bottomrule
        \end{tabular}
        \caption{Approximations of $\Phi_{0;1}^{-1}(x + \Delta x)$}
    \end{table}
    \begin{align*}
        \Phi_{0;1}(x) &= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} e^{- t^2 / 2} \mathrm{d} t &
        \Phi_{0;1}(1.65) &\approx 0.9505\\
        \Phi_{\mu; \sigma^2}(x) &= \Phi_{0;1} \left (\frac{x-\mu}{\sigma} \right ) &
        \Phi_{0;1}(-x) &= 1 - \Phi_{0;1}(x)\\
        z_\alpha &= \Phi_{0;1}^{-1}(1-\alpha)
    \end{align*}
\end{document}