# References on (more or less) explicit calculations of probability distributions of nonlinear transformations of random variables

Premise. After a former question and related answer, I searched for references on the calculation of the probability distributions of nonlinear functions of (one or more) random variables, in order to build a firm basis for dealing with engineering problems which require such calculations in a rigorous manner.

The question

What are the works where the methods for calculating probability distributions of nonlinear transformation of random variables occupies a central place?

I listed here a few guidelines in order to the ease answering to the question.

• The works should be abstract in the sense that the results they present should be rigorous and widely applicable (i.e. applicable to the widest range of problems, or abstract the Italian way).
• The "flavor" has to be analytic in the sense that the reference should deal with the problems in a constructive way by using techniques inherited from real and complex analysis.
• The work should aim at being comprehensive, therefore I prefer monographs/textbooks instead of single research works, even if survey papers on the topic are welcome.

Just to give a few examples, following more or less closely the above guidelines, I have identified the following works:

• Deutsch, R. Nonlinear transformations of random processes, (English) Prentice-Hall International Series in Applied Mathematics. Englewood Cliffs, N.J.: Prentice-Hall, Inc., pp. XI+157 (1962), MR0148499, Zbl 0125.36801.
• Rohatgi, V. K., An introduction to probability theory and mathematical statistics (English) Wiley Series in Probability and Mathematical Statistics. New York-Chichester-Brisbane: John Wiley & Sons, a Wiley-Interscience Publication, pp. XIV+684 (1976), MR0407916, Zbl 0354.62001.
• Springer, M. D., The algebra of random variables, (English) Wiley Series in Probability and mathematical Statistics. New York-Chichester-Brisbane: John Wiley & Sons, pp. XIX+470 (1979), MR0519342, Zbl 0399.60002.

I think this book will be very useful for your purpose

Probabilità e informazione

I suggest you to read the whole book but, in particular,

• Chapter 4 for single rv

• Chapter 6 for pairs of rv's

• Chapter 8 for vectors of rv's

• I apologize for the later in accepting your answer. The reference you pointed out is nice, though I hoped for answer citing more specialized monographs: perhaps there's a gap in the literature. Jul 14 '20 at 10:06