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I often find myself asking questions like, "I know this variable $x$ lies in $(0,1)$ and most of the mass lies in $(0,.20)$ and then declines continuously towards 1. What distribution can I use to model it?"

In practice, I wind up using the same few distributions over and over again simply because I know them. Instead, I'd like to look them up in a more systematic way. How do I go about accessing the wealth of work that probabilitists have done developing all of these distributions?

Ideally I'd like a reference organized by properties (region of support, etc.), so I can find distributions by their characteristics and then learn more about each distribution based on the tractability of the pdf/cdf and how closely the theoretical derivation fits the problem I'm working on.

Does such a reference exist, and if not, how do you go about choosing distributions?

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9 Answers

up vote 7 down vote accepted

The most comprehensive collection of distributions and their properties that I know of are

Johnson, Kotz, Balakrishnan: Continuous Univariate Distributions Volume 1 and 2;

Kotz, Johnson, Balakrishnan: Continuous Multivariate Distributions;

Johnson, Kemp, Kotz: Univariate Discrete Distributions;

Johnson, Kotz, Balakrishnan: Multivariate Discrete Distributions;

The books have a broad subject index. All books are from Wiley.

Edit: Oh yes and then there also is the nice poster displaying properties and relationships between univariate distributions. http://www.math.wm.edu/~leemis/2008amstat.pdf This might be of further interest.

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You should find all of them on Google books for a peek. –  Momo Apr 8 '12 at 16:27
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(+1) These are the classical references and a great place to start. I'm also a big fan of the poster, especially when printed at actual poster size. I've seen a few different incarnations of it. –  cardinal Apr 8 '12 at 18:24
    
The poster looks awesome. :-). The books look...intimidating. –  Ari B. Friedman Apr 9 '12 at 12:56
    
@gsk3: The books are desk references. They're intended to be (somewhat) comprehensive. –  cardinal Apr 9 '12 at 13:07
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I think if you got the univar book, drilled a hole through it, mounted it to one end of a pole, and did the same with the multivar book on the other side, you'd have a nice zombie sledgehammer. –  Ari B. Friedman Apr 9 '12 at 15:36
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honestly, there are way too many distributions that I have no idea about. I do believe however that knowing them is not an asset, one must know how to use them. Anyway, back to your question, I always find this diagram quite informative and useful, it's like probability distributions cheatsheet.

http://jonfwilkins.com/wp-content/uploads/2013/06/BaseImage.png

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+1 I was thinking this would be useful: you saved me having to search for the link! –  whuber Feb 8 at 1:11
    
I believe that diagram is originally from a paper in American Statistician. –  Glen_b Feb 8 at 1:18
    
@Gleb_b: You are right, I came across that diagram the other day: math.wm.edu/~leemis/2008amstat.pdf –  Hibernating Feb 8 at 1:30
    
Along the lines of your diagram, I highly recommend this blog post by @JohnD.Cook: Clickable diagram of probability distribution relationships. –  gung Feb 8 at 3:11
    
Wow, +1 for the diagram. First time I've seen it but it is packed full of information. Just don't use that when designing traffic patterns! –  Underminer Feb 8 at 4:09
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Merran Evans, Nicholas Hastings, Brian Peacock - Statistical distributions - John Wiley and Sons

I have the second edition and the distributions are in simple alphabetical order (from Bernoulli to Wishart central distribution).

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No book could cover all distributions, as it is always possible to invent new ones. But

Statistical distributions by Catherine Forbes et al. is a concise book covering many of the more commonly used distributions

while

A primer on statistical distributions by N. Balakrishnan and V.B. Nezvorov

is also fairly concise, but rather more mathematically oriented.

The nearest approach to a treatise is the series started by N.L. Johnson and S. Kotz, being continued by A.W. Kemp and N. Balakrishnan, and currently published by John Wiley.

This isn't a complete list even of surveys of distributions, but Googling your local Amazon site easily gets you other ideas.

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+1 Johnson & Kotz has been a great resource for me for decades, but the price is appalling. It would be nice to find an affordable version. –  whuber Feb 8 at 1:13
    
@whuber A new edition of one of the volumes amazon.com/… is cited for publication in August 2014. Wiley are currently charging more for a copy of the 1994 edition. –  Nick Cox Feb 8 at 1:19
    
Thanks for the links. 'Statistical distributions' seems to be more student friendly book –  umair Feb 8 at 4:11
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This is pretty decent....and FREE!! It covers over 40 distributions from A to Z, with each distribution described with its formulas, moments, moment generating function, characteristic function, how to generate a random variate from this distribution, and much more. Very nice for a free pdf.

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@gung sure thing. I'll do a little more "marketing" for it, although following the link and seeing it will speak for itself. –  Eupraxis1981 Feb 8 at 2:47
    
Thanks for the link. Although this is a free resource, it is hard to understand because everything is explained using maths. Even the text uses maths terminology. –  umair Feb 8 at 4:04
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Ben Bolker's "Ecological Models and Data in R" has a section "bestiary of distributions" (pp 160-181) with descriptions of the properties and applications of many common and useful distributions.

It is written at the level of a grad level course in ecology, so it is accessible to non-statisticians. Less dense than the Johnson, Kotz et al references in the answer by @Momo, but gives more practical details than a list or appendix might.

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The Loss Models by Panjer, Wilmot and Klugman contains a good appendix regarding distribution pdf, their support and parameter estimation.

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A study of bivariate distributions cannot be complete without a sound background knowledge of the univariate distributions, which would naturally form the marginal or conditional distributions. The two encyclopedic volumes by Johnson et al. (1994, 1995) are the most comprehensive texts to date on continuous univariate distributions. Monographs by Ord (1972) and Hastings and Peacock (1975) are worth mentioning, with the latter being a convenient handbook presenting graphs of densities and various relationships between distributions. Another useful compendium is by Patel et al. (1976); Chapters 3 and 4 of Manoukian (1986) present many distributions and relations between them. Extensive collections of illustrations of probability density functions (denoted by p.d.f. hereafter) may be found in Hirano et al. (1983) (105 graphs, each with typically about five curves shown, grouped in 25 families of distributions) and in Patil et al. (1984).

This is from Chapter 0 of a book on continuous bivariate distributions, which provides an elementary introduction and basic details on properties of various univariate distributions. I remember I enjoyed reading Ord (1972) very much, but I can't now remember why.

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The series of books by Johnson, Kotz & Balakrishnan (edit: which Nick has also mentioned; the original books were by the first two authors) are probably the most comprehensive. You probably want to start with Continuous Univariate Distributions, Vols I and II.

A couple more:

Evans, Hastings & Peacock, Statistical Distributions

Wimmer & Altmann, Thesaurus of univariate discrete probability distributions

There's also many other books, sometimes for more specialized applications.

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Evans, Hastings and Peacock is a previous edition of the book now first authored by Catherine Forbes, which I mentioned. Continuous univariate distributions is the exact title. –  Nick Cox Feb 8 at 9:32
    
@Nick Thanks on the title thing. That was a typo resulting from moving words around in edit. Sorry about not spotting I had duplicated that other one. –  Glen_b Feb 9 at 1:51
    
You're welcome. We're all duplicating a previous answer. (I did check first, but did not find it.) –  Nick Cox Feb 9 at 10:40
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