References to Various Probability Models

[Steve W]

This from Luis Edgar Rodriguez Abreu:

I am working with conditional probabilities in the occurrence of earthquakes, and I have found some problems trying to find a good references about the Brownian Passage Time model (inverse Gaussian), do you any good reference of this? 

This is a good question -- precisely the sort of thing that I hope might be of general interest -- so I'll post my response here.

One of the best set of references to the technical details of various probability distributions is the series originally by Johnson and Kotz in the 1970s.  The newer (2nd edition) version of these is now in 5 volumes:

Continuous Multivariate Distributions, Volume 1, Models and Applications, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Continuous Univariate Distributions, Volume 1, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Continuous Univariate Distributions, Volume 2, 2nd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Discrete Multivariate Distributions by Samuel Kotz, N. Balakrishnan and Normal L. Johnson
Univariate Discrete Distributions, 3rd Edition by Samuel Kotz, N. Balakrishnan and Normal L. Johnson

Probably the most useful is continuous univariate distributions (Vol 1), which includes the inverse Gaussian as a separate chapter.  There's also lots of good material in its first chapter on the basics of (continuous) PDFs and their properties.  Some notes:

* I'm only familiar with the first version, which dates from the 1970s (and which I found an excellent resource).  I have no experience with the 2nd edition, but some comments I find online suggests that they may be a bit less reliable (readers mention typos).  

* These books were (and are) rather pricey.  I hope they would be accessible to you via some research library.  

For your general interest, the "inverse Gaussian" distribution describes the time when "Brownian motion" (a nonstationary process whose mean and variance both increase linearly with time) crosses a prescribed boundary.  It is used in lots of cumulative stress/damage models, such as fatigue of metals.

-- Steve

[Luis RA]

Hello Steve, 

Thanks a lot for the references, I will take a look. Also I think it was good to post the question here, if somebody else is interested to share some other references is more than wellcome. 

Cheers, 

Luis