- Virtual Laboratories
- Introduction

The goal of this project is to provide a set of free, high quality, web-based resources for students and teachers of probability and statistics. 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 is divided into three major parts:

- Expository Material
- Ancillary Material
- The Object Library.

All ancillary materials open in small pop-up windows. 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.

This project uses a number of advanced web technologies. Almost all of these are open, published standards endorsed by the World Wide Web Consortium, the definitive standards group for the web. 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.

- XML. Most of the files that make up this project are in XML format, which stands for eXtensible Markup Language. XML is a general markup language used to construct specialized markup languages.
- XHTML.The basic expository text is written in XHTML, which stands for eXtensible Hypertext Markup Language (a particular type of XML). XHTML is the rigorous, XML version of ordinary HTML.
- MathML. Mathematical expressions are written in MathML, which stands for Mathematics Markup Language (a particular type of XML). MathML comes in two versions--Presentation MathML encodes mathematical notation in a precise way, while Content MathML encodes the actual structure of the mathematics. Mathematical expressions encoded in MathML are rendered in correct notation and can be exported into other MathML-enabled applications for further processing. Our project uses an extended version of Content MathML, customized for probability and statistics.
- Java. Most applets are written in Java, a platform-independent programming language widely used for embedding programs in web pages.
- JavaScript. Some applets are written in JavaScript, and in general the ancillary elements (applets, data sets, and biographical sketches) are opened in small child windows that are launched with JavaScript. The standard version of Java Script is actually known as ECMAScript.
- CSS. The presentation of most XHTML elements is controlled by CSS, which stands for Cascading Style Sheet.
- XSL. The translation of Content MathML to Presentation MathML is accomplished by XSL, which stands for eXtensible Style Language (a particular type of XML).
- PNG. Graphics files are in PNG format, which stands for Portable Network Graphics. This is a format for compressed graphics files with no loss of information, and is similar to GIF format, but better and without the patent encumbrances of GIF.

To use this project properly, you will need the latest version of the Mozilla Firefox browser, with the appropriate fonts installed. You will also need the latest version of the Java plug-in, and you must have JavaScript enabled. Firefox is currently the only browser that supports all of the open standards needed for our project. Click on the appropriate links, or on the icons below, to install or update your browser and to install the plug-ins.

If you use a browser other than Firefox, you should be able to access the pages and run the applets. However, mathematical expressions will probably not display correctly, and features that rely on advanced CSS (such as exercise numbers) may not display correctly. The list below discusses the capabilities of some common browsers in more detail.

- The Opera browser renders Presentation MathML using CSS but does not support Content MathML. Opera supports all other technologies used in this project.
- Internet Explorer (on Windows) supports Presentation MathML via the MathPlayer plug-in, but not the extended version of Content MathML used in this project. Moreover, IE has the worst support for CSS among all browsers, so other elements of this project will not display properly. For example, exercise numbers, which are automatically generated via CSS will not display.
- Most other common browsers, such as Chrome and Safari, do not support MathML, but do support the other technologies. You may be able to make sense of simple mathematical expressions, but not complicated ones.

The sections of expository text in the main chapters are provided in PDF form, in case you are unable to use Firefox or if you need to print the text. However, interactive features such as hyperlinks and applets are not available in the PDF pages.

The expository material is divided into chapters, similar to chapters in a conventional book. The chapters explore the basic theory and applications of probability, statistics, and certain special models and random 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. A few sections contain advanced ideas involving measure theory or analysis, but these sections are identified and can be omitted if necessary.

As we have already noted, the expository material is mostly in the form of exercises. These are of three basic types, each keyed with a special icon.

- Mathematical exercises require only paper and pencil. Many of these develop the basic mathematical theory and thus involve proofs or derivations (broken into small steps). Other mathematical exercises are computational or involve examples and special cases. It is very important to read the exercises that develop the basic theory, even if you are unable to work all of them.
- Simulation exercises are based on applets (described below). In some cases, the student is only asked to observe the behavior of a simulation or generate and observe data. In other cases, hand computations may also be required. In some cases, a calculator or statistical software will be necessary. In all cases, the applet exercises are intended to demonstrate the mathematical theory in a dynamic and interactive way.
- Data analysis exercises are keyed to data sets. These exercises require statistical software or at least a calculator.

Answers to many of the computational exercises are provided via a pop-up window. Exercises with answers are keyed with the icon.

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

The applets generally fall into two basic types:

- Simulation applets. These are simulations of random processes, designed to show the agreement between the predictions of the mathematical theory and the behavior of the process. These are generally referred to as experiments in the text. Generally, the student can vary parameters and choose among various probability distributions that drive the simulation.
- Data generation applets. These are applets in which the student generates the data, by making choices in a game, or clicking in a scatterplot or number line. These are generally referred to as games in the text.

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 applets with little or no instructions. The applets run in a separate window, so that the student can easily move back and forth between the applets and the corresponding hypertext discussion, and so that a student can keep an applet open and running as she browses through the text. The following image is a screen shot the Dice Experiment. You can click on the image to run the experiment

The applet 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 applet objects have tool tips, small pop-up boxes that explain the object. Rest the cursor on an object to display the tool tip.

Applets that are simulations of random processes all have the Main Toolbar shown below:

The Main Toolbar has the following basic buttons and controls:

- Step. This button runs the experiment one time, updating data and graphs. In some cases sound and delays are incorporated to give the student maximum feedback about the random process.
- Run. This button runs the experiment repeatedly. The frequency of data and graphical updates can be controlled by the student. No sounds or delays are incorporated so that the simulation will run fast.
- Stop. This button stops the simulation but preserves all data and graphs.
- Reset. This button clears all simulation data and graphs and restores the random process to its initial state.
- Information. This button displays basic copyright and support information about the applet.

The update frequency is selected from the first list box on the main toolbar. This number determines how often graphical and numerical displays are updated in run mode. In most applets, you can select an update frequency of 1, 10, 100, or 1000. In some applets, other update rules are provided. An update occurs automatically if the simulation is stopped by clicking the stop button.

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 applets you can select a stop frequency of 10, 100, 1000, or 10000. In some applets, other stop rules are provided.

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 applet window, below the main toolbar. All toolbars can be undocked

into separate, small windows which can then be moved to a convenient place on the desktop. In particular, if some of the controls on a toolbar are not visible, undock the toolbar to reveal these controls. The following image shows the parameter toolbar of the Dice Experiment.

Our project has a collection of approximately 10 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, a link to the data set in the standard tab-separated text format is provided, so that the data set can be downloaded and opened in a statistical or spreadsheet program.

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

Our project includes tables for several common distributions, including the standard normal distribution, the student `t` distribution, and the chi-square distribution.

The objects

that make up this project 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 all of these objects and instructions for downloading and using them.

In particular, the applets in this project are intended to stand alone, as virtual versions of random processes and statistical games. In turn, the applets are built out of reusable components, according to a carefully designed object model. All of the applets and components of applets (both in executable form and source files) are freely available in the object library. The objects in the library are divided into three *packages*:

- Devices. This package contains objects with visible interfaces, including virtual version of physical objects (such as coins, dice, balls, and spinners), and custom interface objects (such as tables and graphs).
- Distributions. This package contains mathematical objects such as probability distributions and data structures.
- Experiments. The objects in this package correspond to the applets themselves. An applet can be downloaded and dropped into a user's web page along with other elements of her design (text, data sets, etc.). No programming knowledge is required to use the applets in this way.

A user with programming experience can download components and build a custom applet in a fraction of the time required to program the applet from scratch.

The object library also has descriptions and links to other reusable objects, including

- The CSS screen and print style sheets
- The XSL file for translation of Content MathML to Presentation MathML
- The JavaScript files

A you are here

navigational map is given on each page. The contents page of a chapter has links the virtual laboratories home page and to the contents pages of the other chapters. Each section in a chapter has links to the 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:

- Table of contents (site map)
- External Resources
- List of keywords
- Feedback form
- Information on rights and permissions

Our site has links to a number of related sites in probability and statistics; these links are colored dark blue. Pages from external sites are opened in a new browser window.

The following list gives the meanings of the icons that are used frequently in this project:

- represents a mathematical exercise.
- represents a link to the answer for a computational exercise.
- represents the answer to a computational exercise.
- represents an applet or a simulation exercise.
- represents a data set or a data analysis exercise.
- represents a programmatic object from the object library, in an abstract sense.
- represents a compressed Java Archive (JAR) file that contains all class and resource files for an object in the library.
- represents a compressed ZIP file that contains all source and resource files for an object in the library.
- represents a person or a biographical sketch.
- represents a distribution table.
- represents a quote.
- represents a book.
- represents an article.
- represents an external website.