1. Random
  2. Introduction



Random (formerly Virtual Laboratories in Probability and Statistics) is a website devoted to probability, mathematical statistics, and stochastic processes, and is intended for teachers and students of these subjects. Many of the elements are designed so that they can be modified and reused to fit the needs of students at various levels. The project has two basic types of resources:

Ancillary materials open in small pop-up windows on laptop and desktop computers, and in separate tabs on tablets and phones. Thus, these materials can remain open and accessible while you browse the expository material in the main window. The expository material has extensive links to the ancillary materials, but some of these materials are also designed to stand alone so that they can be used in other projects. The ancillary materials are of the following types:

The major components of the project are discussed in more detail below.

Technologies and Browser Requirements

This project uses HTML5 web technologies. Almost all of these tecnologies are open, published standards endorsed by various standards groups, including the World Wide Web Consortium. One of our goals is for this site to be fully standards compliant and to implement best practices for web-based expository mathematics. The list below describes the main technologies used; follow the links for more information.

Mathematical Notation

Display of mathematical notation is handled by the open source MathJax project (which in turn uses JavaScript and CSS). Mathematical expressions rendered by MathJax scale appropriately with the surrounding text, and can be copied and pasted into other applications.


To use this project properly, you will need a modern browser that supports these technologies. The latest versions of Chrome, Firefox, Opera, and Safari are the best choices. The Internet Explorer and Edge browsers for Windows do not fully support the technologies used in this project.

Expository Material

The expository material is divided into chapters, similar to chapters in a conventional book. The chapters explore the basic theory and applications of probability, mathematical statistics, and certain special models and stochastic processes. Each chapter in turn is divided into web pages, similar to sections in a conventional book. Each section explores a particular topic, mostly through a series of exercises that guide the student through the development of the mathematical theory and the development of probabilistic intuition.


The expository text assumes knowledge of calculus, at the standard undergraduate level. A few sections require knowledge of linear algebra, at the standard undergraduate level. No prior knowledge of probability or statistics is assumed. Many of the chapters are divided into basic topics that are essential to all students, and advanced and special topics that can be omitted for new students.


The exercises in the expository sections are of three basic types, each keyed with a special icon.

Proofs of most theoretical exercises and answers for most computational exercises are provided. On supported browsers (Chrome, Firefox, Opera, and Safari), you can expand and contract the proofs and answers. At the top of each expository page (in the navigation line, on the right), two buttons are provided that allow you to expand or contract all details (proofs and answers) on that page.


The web apps are designed to demonstrate the mathematical theory in a dynamic, interactive way. Each app runs in a separate window, with the small amount of text necessary to describe the app and its notation, but with little explicit mathematical exposition. Thus, the apps can be used with the expository portion of this project, or by themselves, at various mathematical levels.

The apps generally fall into two basic types:

Graphical User Interface

A standard Graphical User Interface (GUI) is used, with command buttons, scroll bars and list boxes. There is no programming or command language, so students should be able to run the apps with little or no instructions. The apps run in a separate window, so that the student can easily move back and forth between the apps and the corresponding exposition, and so that a student can keep an app open and running as she browses through the text.

The app output is displayed numerically and graphically in a set of coordinated tables and graphs. A consistent color-coding is used. Graphical objects that depend only on the distributions or parameters are shown in blue, while graphical objects that depend on data (either simulation or student generated) are shown in red. Most app objects have tool tips, small pop-up boxes that explain the object. Rest the cursor on an object to display the tool tip.

The Main Toolbar

Apps that are simulations of random processes all have the Main Toolbar with the following basic buttons and controls:

The stop frequency is selected from the second list box on the main toolbar. The stop frequency is the number of runs before the simulation stops in run mode. In most apps you can select a stop frequency of 10, 100, 1000, or 10000. In some apps, other stop rules are provided.

Parameter Toolbars

The student can easily vary the parameters, select distributions, and choose among appropriate modeling assumptions using list boxes, scroll bars, and pop-up dialog boxes. These controls appear on parameter toolbars at the top of the app window, below the main toolbar.

The Object Library

The objects that make up this project, particularly the interactive web apps, are designed, to the extent possible, so that they can be re-used in other projects and modified, if necessary. The Object Library contains descriptions of of these objects and instructions for using them.

Other Ancillary Materials

Data Sets

Our project has a number of data sets from real statistical studies, many of them historically interesting. The page for each data set has a brief description of the data set and its source. The data set itself is given either in a table or in a scrollable text area, so that the data can be copied and pasted into a statistical or spreadsheet program.. Also, in most cases, links to the data set in standard tab-separated text format and comma-separated text format are provided, so that the data set can be downloaded and opened in a statistical or spreadsheet program.

Biographical Sketches

A brief biographical sketch is provided for most of the famous persons referenced in the expository material. There are approximately 100 biographical sketches in total.


This is a small collection of paintings and other art forms strongly influenced by games of chance or randomness generally.



A you are here navigational map is given on each page. The contents page of a chapter has links to the Random home page and to the contents pages of the other chapters. Each section in a chapter has links to the Random home page, to the contents page of the chapter and to the other sections in the chapter. Links that open in the main browser window are colored blue.

Additionally, the footer on each page has links to the ancillary materials that open in separate, small windows without menus or toolbars. Links to ancillary materials are colored red. Each footer also has links to the following: