About the chart:

The list on the left-hand side displays the names of the 76 probability distributions (19 discrete distributions given by the rectangular boxes and 57 continuous distributions given by the rectangular boxes with the rounded corners) present in the chart. Hovering your mouse over the name of a distribution highlights the distribution on the chart, along with its related distributions. Depending on the size of your browser window, you might have to adjust the display to find the distribution you are looking for. You may scroll the chart window or zoom in and out with the + and - buttons as needed.

Each distribution on the chart, when clicked, links to a document showing detailed information about the distribution, including alternate functional forms of the distribution and the distribution's mean, variance, skewness, and kurtosis.

What is a univariate distribution?

A univariate probability distribution is used to assign a probability to various outcomes of a random experiment. A random experiment is one whose outcome can not be predicted with certainty prior to conducting the experiment. When the set of all possible outcomes to a random experiment is countable or a countable infinity, the probability distribution can be described by a probability mass function and the associated random variable is discrete. Otherwise, the probability distribution can be described by a probability density function and the associated random variable is continuous. A mix of these two cases is known as a mixed discrete-continuous distribution. Illustrations of a probability mass function in the case of rolling a pair of fair dice and summing the outcomes on the up faces and a probability density function in the case of the well-known normal distribution can be seen by clicking here.

A univariate probability distribution is the probability distribution of a single random variable. This is in contrast to a bivariate or multivariate probability distribution, which defines the probability distribution of two or more random variables.

What do the arrows mean?

Solid lines represent special cases and transformations from one distribution to another.

Dashed arrows are used for asymptotic relationships, typically as the limit as one or more parameters approach the boundary of the parameter space.

Dotted arrows represent Bayesian relationships.

Upon interacting with the chart, outbound arrows are highlighted in yellow pointing away from the selected distribution. Incoming arrows and related distributions are highlighted in white. Placing the cursor over an arrow turns the arrow blue. Clicking the arrow reveals a .pdf file that contains a proof when one exists. The accompanying transformation or parameterization will be highlighted next to the arrow.

What do the letters just below the distribution names indicate?

Placing the cursor over a letter for a property turns the letter blue. Clicking the property reveals a .pdf file that contains a proof when one exists.

What is the meaning of the parameters associated with the univariate probability distributions?

Parameters are used to enhance the flexibility of a univariate probability distribution. The normal distribution with its bell-shaped probability density function, for example, might be an appropriate probability model the annual return for a stock index or the diameter of a ball bearing by adjusting the values of its parameters.

Generally speaking, there are three types of parameters associated with a continuous distribution. A location parmeter shifts the probability density function to the left or to the right along the horizontal axis. A scale parameter contracts or expands the scale associated with the horizontal axis of the probability density function. A shape parameter changes the shape of the probability density function. An example of a location parameter is the mean of a normal random variable; an example of a scale parameter is the standard deviation of a normal random variable; an example of a shape parameter is the degrees of freedom of a t random variable.

Are there errors on the chart?

Yes. The chart is basically identical to that which was published in The American Statistician. In writing the proofs for some of the properties and relationships, we have uncovered errors. In addition, we were unable to complete some of the proofs. They are listed by categories below.

Are there other univariate distributions not on the chart?

Yes. We were not able to squeeze all of them onto the chart. Here is a partial list of distributions that are not among the 76 given on the chart:

Are there other relationships between univariate distributions?

Some relationships did not fit on the chart because the chart needed to be a planar graph. Other relationships involve the combination of two random variables to create a third. Examples include

Is there more information available?

Clicking on a distributon's name will download a .pdf file that includes the cumulative distribution function, survivor function, hazard function, cumulative hazard function, inverse distribution function, and (where applicable) the moments and moment generating function. For even more information, see the "Links" tab. A nice on-line compendium is given by Gavin Crooks

Who developed the interactive graphic website?

The main developers were Lawrence Leemis, Daniel Luckett, Austin Powell, Jackie Taber, and Peter Vermeer (all from the COR (Computational Operations Research) program at The College of William & Mary). Other contributors are: Tim Adams, Stefano Bettelli, Ruben Becerril Borja, Hans Brunner, Ryan Carpenter, Erin Catlett, Danny Cogut, Chad Conrad, Laura Decena, Evan Elsaesser, Luke Godwin--Jones, Matt Goldman, Matt Hanson, Yan Hao, Billy Kaczynski, Andrew Mashchak, Maximilian Matthe, James McCormack, Lee McDaniel, Lauren Merrill, Erik Olsen, Scott Percic, Jean Peyhardi, Robin Ryder, Evan Saltzman, Bruce Schmeiser, Christian Schröder, Erich Schubert, Witold Siedlaczek, Maciej Swat, Chris Thoma, Melissa Tilashalski, Erik Vargo, Chris Weld, Jeff Yang, Vincent Yannello, Hang Yu, and Andrei Zorine. Support also came from the National Science Foundation.

What if I found something wrong, something missing, or want to contribute a proof?

Questions and concerns may be directed to leemis (AT) math (doTT) wm (d0T) edu