A theorem for physicists in the theory of random variables
Abstract
A theorem is presented which tells how to calculate the joint probability distribution of m random variables that have been defined as functional transformations of n given random variables (m, n≥1). Although the theorem involves Dirac delta functions and therefore has a rather formal appearance, it turns out to be surprisingly useful. It is used here to develop a number of important results in random variable theory, including: the central limit theorem, the density functions of some common probability distributions (lognormal, chisquare, and Student'st), the imposition of constraints on random variables, some sampling properties of the normal distribution, and the chisquare goodnessoffit test. Special attention is given to aspects of these topics that find fruitful application in physics. A broad conclusion seems to be that the theorem presented here provides a systematic way of obtaining many results in random variable theory which, although quite useful in physics, are normally found derived only in moderately advanced mathematics textbooks.
 Publication:

American Journal of Physics
 Pub Date:
 June 1983
 DOI:
 10.1119/1.13221
 Bibcode:
 1983AmJPh..51..520G
 Keywords:

 02.50.Cw;
 Probability theory