Probability Density Function - Dependent Variables and Change of Variables

Dependent Variables and Change of Variables

If the probability density function of a random variable X is given as f_X(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable Y = g(X). This is also called a “change of variable” and is in practice used to generate a random variable of arbitrary shape f_g(X) = f_Y using a known (for instance uniform) random number generator.

If the function g is monotonic, then the resulting density function is

Here g−1 denotes the inverse function.

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,

or

For functions which are not monotonic the probability density function for y is

where n(y) is the number of solutions in x for the equation g(x) = y, and g−1_k(y) are these solutions.

It is tempting to think that in order to find the expected value E(g(X)) one must first find the probability density f_g(X) of the new random variable Y = g(X). However, rather than computing

one may find instead

The values of the two integrals are the same in all cases in which both X and g(X) actually have probability density functions. It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former.