Drawing cumulative distribution function for a discrete variable

Question

Can you help me out with drawing a simple cumulative distribution function of a discrete variable, which has the following values:

x=1, f(x)=1/15;
x=2, f(x)=2/15;
x=3, f(x)=1/5;
x=4, f(x)=4/15;
x=5, f(x)=1/3

Most resources show how to do it for continuous variables. The question is very trivial because I am a newbie. Thank you.

EDIT:

\documentclass[paper=a4, fontsize=11pt]{scrartcl} % A4 paper and 11pt font size

\usepackage[T1]{fontenc} % Use 8-bit encoding that has 256 glyphs
\usepackage{fourier} % Use the Adobe Utopia font for the document - comment this line to return to the LaTeX default
\usepackage[english]{babel} % English language/hyphenation
\usepackage{amsmath,amsfonts,amsthm} % Math packages

\usepackage{lipsum} % Used for inserting dummy 'Lorem ipsum' text into the template

\usepackage{sectsty} % Allows customizing section commands
\allsectionsfont{\centering \normalfont\scshape} % Make all sections centered, the default font and small caps

\usepackage{fancyhdr} % Custom headers and footers
\usepackage{xfrac}
\pagestyle{fancyplain} % Makes all pages in the document conform to the custom headers and footers
\fancyhead{} % No page header - if you want one, create it in the same way as the footers below
\fancyfoot[L]{} % Empty left footer
\fancyfoot[C]{} % Empty center footer
\fancyfoot[R]{\thepage} % Page numbering for right footer
\renewcommand{\headrulewidth}{0pt} % Remove header underlines
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\setlength{\headheight}{13.6pt} % Customize the height of the header

\numberwithin{equation}{section} % Number equations within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)
\numberwithin{figure}{section} % Number figures within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)
\numberwithin{table}{section} % Number tables within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)

\setlength\parindent{0pt} % Removes all indentation from paragraphs - comment this line for an assignment with lots of text

%----------------------------------------------------------------------------------------
%   TITLE SECTION
%----------------------------------------------------------------------------------------

\newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height

\title{ 
\normalfont \normalsize 
\textsc{\LARGE XXXX} \\ [20pt]
\textsc{ XXX}  \\ % Your university, school and/or department name(s)
\textsc{ XXX}  \\ [20pt]
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule
\huge Problem Set 1 \\ % The assignment title
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule
}

\author{XXX  \textsc{XXX}} % Your name

\date{\normalsize\today} % Today's date or a custom date

\begin{document}

\maketitle % Print the title

%----------------------------------------------------------------------------------------
%   PROBLEM 1
%----------------------------------------------------------------------------------------


1. Let X be a discrete random variable with density function


\[ P(x) = \left\{ 
  \begin{array}{l l}
    cx & \quad \text{for $x$ =1, 2, 3, 4, 5}\\
    0 & \quad \text{otherwise}
  \end{array} \right.\]
 \\ The discrete probability density function (PDF) of a discrete random variable $X$ provides the probabilities P($X=x$) for all possible values of $x$. In our case, \\

P($x$=1)=$1\times c$,\\
P($x$=2)=$2\times c$,\\
P($x$=3)=$3\times c$,\\
P($x$=4)=$4\times c$,\\
P($x$=5)=$5\times c$, $0$ otherwise\\

$1\times c +   2\times c +3\times c +4\times c +5\times c=1 \rightarrow c=\dfrac{1}{15}$

  The probability density function for $X$ is:\\

\begin{tabular*}{1\textwidth}{@{\extracolsep{\fill} } | c | c | c | c | c | c |} 
  \hline
   x & 1 & 2 & 3 & 4& 5 \\ [1ex]
  \hline
  \raisebox{2ex}f(x)& $1\times \dfrac{1}{15}=\dfrac{1}{15}$  & $2\times \dfrac{1}{15}=\dfrac{2}{15}$   & $3\times\dfrac{1}{15} =\dfrac{1}{5}$  & $4\times \dfrac{1}{15}=\dfrac{4}{15}$  & $5\times \dfrac{1}{15} =\dfrac{1}{3}$ \\ [1.5ex]
  \hline
\end{tabular*}\\\\

 A. What is $c$? \\\\
 $c=\dfrac{1}{15}$ \\\\

 B. Find P($X$ is odd).\\\\
 P($X$ is odd)$= P($X$ =1)+ P($X$ =3)+ P($X$ =5)=\dfrac{1}{15}+\dfrac{1}{5}+\dfrac{1}{3}=\dfrac{9}{15}$ \\\\


 C. What is the cumulative distribution function for X? Plot the function.\\\\

 $P(X\leq1)=\dfrac{1}{15}$\\\\
 $P(X\leq2)=\dfrac{1}{15}+\dfrac{2}{15}=\dfrac{1}{5}$\\\\
 $P(X\leq3)=\dfrac{1}{15}+\dfrac{2}{15}+\dfrac{1}{5}=\dfrac{6}{15}$\\\\
 $P(X\leq4)=\dfrac{1}{15}+\dfrac{2}{15}+\dfrac{1}{5}+\dfrac{4}{15}=\dfrac{2}{3}$\\\\
 $P(X\leq5)=\dfrac{1}{15}+\dfrac{2}{15}+\dfrac{1}{5}+\dfrac{4}{15}+\dfrac{1}{3}=1$\\\\
\\\\
\\\\
\\\\
\\\\ 
\\\\
\\\\

\end{document}

Answer 1

You can use the approach from Probability density function of Uniform Distribution.

The actual plot is defined using this code:

\begin{axis}[
    clip=false,
    jump mark left,
    ymin=0,ymax=1,
    xmin=0, xmax=6,
    every axis plot/.style={very thick},
    discontinuous,
    table/create on use/cumulative distribution/.style={
        create col/expr={\pgfmathaccuma + \thisrow{f(x)}}   
    }
]
\addplot [red] table [y=cumulative distribution]{
x f(x)
0 0
1 1/15
2 2/15
3 1/5
4 4/15
5 1/3
6 0
};
\end{axis}

---

Here's the complete example document.

The big chunk of code between \makeatletter and \makeatother provides the discontinous plot style. The code chunk can be put in your preamble, it doesn't need to be repeated if you need several of these plots.

\documentclass{article}
\usepackage{pgfplots, pgfplotstable}
\usepackage{amsmath}

\makeatletter
\long\def\ifnodedefined#1#2#3{%
    \@ifundefined{pgf@sh@ns@#1}{#3}{#2}%
}

\pgfplotsset{
    discontinuous/.style={
    scatter,
    scatter/@pre marker code/.code={
        \ifnodedefined{marker}{
            \pgfpointdiff{\pgfpointanchor{marker}{center}}%
             {\pgfpoint{0}{0}}%
             \ifdim\pgf@y>0pt
                \tikzset{options/.style={mark=*, fill=white}}
                \draw [densely dashed] (marker-|0,0) -- (0,0);
                \draw plot [mark=*] coordinates {(marker-|0,0)};
             \else
                \tikzset{options/.style={mark=none}}
             \fi
        }{
            \tikzset{options/.style={mark=none}}        
        }
        \coordinate (marker) at (0,0);
        \begin{scope}[options]
    },
    scatter/@post marker code/.code={\end{scope}}
    }
}

\makeatother

\begin{document}
C. What is the cumulative distribution function for X? Plot the function.

\begin{align*}
P(X\leq1) &= \dfrac{1}{15}\\
P(X\leq2) &= \dfrac{1}{15}+\dfrac{2}{15}=\dfrac{1}{5}\\
P(X\leq3) &= \dfrac{1}{15}+\dfrac{2}{15}+\dfrac{1}{5}=\dfrac{6}{15}\\
P(X\leq4) &= \dfrac{1}{15}+\dfrac{2}{15}+\dfrac{1}{5}+\dfrac{4}{15}=\dfrac{2}{3}\\
P(X\leq5) &= \dfrac{1}{15}+\dfrac{2}{15}+\dfrac{1}{5}+\dfrac{4}{15}+\dfrac{1}{3}=1
\end{align*}

{\centering
\begin{tikzpicture}
\begin{axis}[
    clip=false,
    jump mark left,
    ymin=0,ymax=1,
    xmin=0, xmax=5,
    every axis plot/.style={very thick},
    discontinuous,
    table/create on use/cumulative distribution/.style={
        create col/expr={\pgfmathaccuma + \thisrow{f(x)}}   
    }
]
\addplot [red] table [y=cumulative distribution]{
x f(x)
0 1/15
1 2/15
2 1/5
3 4/15
4 1/3
5 0
};
\end{axis}
\end{tikzpicture}
\par}
\end{document}

---

To center the image, you can wrap the \begin{tikzpicture} ... \end{tikzpicture} block in

{\centering
...
\par}