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, chi-square, and Student's-t), the imposition of constraints on random variables, some sampling properties of the normal distribution, and the chi-square goodness-of-fit 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.