Random variables are sometimes defined as functions of others. For example, parameter estimates are usually functions of the values in a random sample.
In this chapter, we will describe some general methods for finding the distribution of a random variable that is defined as a function of a single variable, Y = g(X).
We start with some general methods to find the distribution of a transformed random variable. One application of these methods shows how to generate random values from an arbitrary distribution from values that have been randomly generated with a Rectangular(0, 1) distribution.
Linear transformations are particularly important in statistics, and formulae are given for the mean and variance of a linear function of a variable. In some families of distribution, linear transformations result in distributions from the same family. In particular, a linear transformation of a normally distributed variable also has a normal distribution.
Finally, if the distribution of a nonlinear transformation of a variable is complex, the delta method may provide approximate values for its mean and variance.