The coverage is quite good for routine, and some non-routine purposes. I find the characteristic functions especially helpful. Each distribution's description of how it arises is also very useful - it's the kind of information that a practitioner needs in order to apply distributions to problems in meaningful ways.
I know that no book can say everything, but a few additions would have improved this book significantly. More discussion of applications would have helped. So would a discussion of general techniques for generating random numbers - inverse distributions, rejection, etc.
The two real weaknesses I found were in the extreme value and the empirical distributions. Extreme values don't stand alone. They often arise in ways dependent on other distributions. An extreme value distribution might describe the results of many experiments that find the largest of N values drawn from distribution P - with different results according to P. These distributions don't have convenient closed forms, but are amenable to some kinds of analysis anyway.
Perhaps the authors do a reasonable job of empirical distributions in the continuous case, but discrete (categorical) cases arise more in my work. Discrete distributions must answer such questions as: given that my sampling may not have found objects of all possible types, how many unknown types are probably still out there? Lots of problems have distributions too complicated for analysis or too poorly understood for book formulas to work, and must be handled empirically. More discussion of empirical techniques would make this a much stronger reference.
Despite its soft spots, this is a very practical reference. I expect it to be a productive member of my technical library.
The book should be seen purely as a handbook on statistical distributions, not as a theoretical reference. The book is ideal for those who make use of statistical distributions in other fields, and who are not necessarily statisticians themselves. I have no formal statistics training, but use distributions extensively in my own work, and found this book very easy to understand. I have been using Johnson and Kotz monographs fairly extensively as references for the distributions in which I am interested, but find this book a much simpler reference for the basic facts of the distributions. In addition, its consistent use of notation across the chapters makes it much easier for the reader to cross reference.
I refrain from giving 5 stars to the book because of a few weaknesses, primarily omissions. Firstly, as an earlier reviewer pointed out, the lack of an index is a little annoying sometimes. Secondly, the bibliography is very slim, and so the reader interested in finding further details, proofs etc., is given very little direction. Thirdly, there are a few obvious omissions, such as the cumulative distribution function for the chi-squared distribution. Fourthly, random number generation is described only when the generation is relatively simple (for example, a method for generating random variates from a gamma distribution is described only for special cases). Finally, I would like to have seen more guidance provided in the sections on parameter estimation, such as first and second derivatives of log-likelihood functions when the estimates have to be derived iteratively.