How to modify density function's plot

Question

I'm trying to reproduce this density function graph

To plot this, I took a cue from this answer, but there are some things that I had modified.

 \documentclass{article}
    \usepackage{tikz}
    \usetikzlibrary{patterns,intersections}
    \usepackage{pgfplots}
    \begin{document}
    \usepgfplotslibrary{fillbetween}
    \begin{tikzpicture}
    \tikzset{
        hatch distance/.store in=\hatchdistance,
        hatch distance=10pt,
        hatch thickness/.store in=\hatchthickness,
        hatch thickness=2pt
    }
\makeatletter
\pgfdeclarepatternformonly[\hatchdistance,\hatchthickness]{flexible hatch}
{\pgfqpoint{0pt}{0pt}}
{\pgfqpoint{\hatchdistance}{\hatchdistance}}
{\pgfpoint{\hatchdistance-1pt}{\hatchdistance-1pt}}%
{
    \pgfsetcolor{\tikz@pattern@color}
    \pgfsetlinewidth{\hatchthickness}
    \pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
    \pgfpathlineto{\pgfqpoint{\hatchdistance}{\hatchdistance}}
    \pgfusepath{stroke}
}
\makeatother


\begin{axis}[
        xmin=-4,xmax=4,
        xlabel={z},
        ymin=0,ymax=1,
        axis on top,
        legend style={legend cell align=right,legend plot pos=right}]
\addplot[name path=A,color=red,domain=-4:4,samples=100] {1/sqrt(2*pi)*exp(-x^2/2)};

\path[name path=B] (axis cs:\pgfkeysvalueof{/pgfplots/xmin},0) -- (axis cs:\pgfkeysvalueof{/pgfplots/xmax},0);

\addplot[pattern=flexible hatch,pattern color=cyan,draw=blue,hatch distance=5pt, hatch thickness=0.5pt]
fill between[
    of=A and B,
    soft clip={domain=-1:1},
];
\end{axis}

\end{tikzpicture}
\end{document}

I don't know how to add a, b, $f_X$, $p(a \le X \le b)$ to the axis.

Answer 1

This can be done by using the xtick and xticklabels options of the axis to set the labels a and b to the x-coordinates -1 and 1. As well as using(axis cs:x, y) to specify a point (x, y) on the plot to draw at in tikz.

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\usetikzlibrary{patterns,intersections}
\usepgfplotslibrary{fillbetween}
\begin{document}
    \begin{tikzpicture}
    \tikzset{
        hatch distance/.store in=\hatchdistance,
        hatch distance=10pt,
        hatch thickness/.store in=\hatchthickness,
        hatch thickness=2pt
    }
\makeatletter
\pgfdeclarepatternformonly[\hatchdistance,\hatchthickness]{flexible hatch}
{\pgfqpoint{0pt}{0pt}}
{\pgfqpoint{\hatchdistance}{\hatchdistance}}
{\pgfpoint{\hatchdistance-1pt}{\hatchdistance-1pt}}%
{
    \pgfsetcolor{\tikz@pattern@color}
    \pgfsetlinewidth{\hatchthickness}
    \pgfpathmoveto{\pgfqpoint{0pt}{0pt}}
    \pgfpathlineto{\pgfqpoint{\hatchdistance}{\hatchdistance}}
    \pgfusepath{stroke}
}
\makeatother


\begin{axis}[
            xmin=-4,xmax=4,
            xlabel={(z)},
            ymin=0,ymax=1,
            axis on top,
            legend style={legend cell align=right,legend plot pos=right},
            xtick={-1,1},
            xticklabels={(a),(b)},
        ]
    \addplot[name path=A,color=red,domain=-4:4,samples=100] {1/sqrt(2*pi)*exp(-x^2/2)};

    \path[name path=B] (axis cs:\pgfkeysvalueof{/pgfplots/xmin},0) -- (axis cs:\pgfkeysvalueof{/pgfplots/xmax},0);

    \addplot[pattern=flexible hatch,pattern color=cyan,draw=blue,hatch distance=5pt, hatch thickness=0.5pt]
    fill between[
        of=A and B,
        soft clip={domain=-1:1},
    ];

    % Add labels
    \draw[->] (axis cs:0, 0.2) to[out=60, in=200] (axis cs:1.2, 0.5) node[right] {\(P(a \le X \le b)\)};
    \node at (axis cs:0.2, 0.45) {\(f_x\)};

\end{axis}

\end{tikzpicture}
\end{document}