Generating Functions

Generating functions are slightly strange-looking objects at first, but they turn out to be extremely useful.

Essentially, these functions package information about a random variable into a power series, whose coefficients or derivatives contain the things we are interested in: namely, probabilities or moments. We then think of the power series as a function which we can manipulate, rather than working with all of these quantities separately.

We begin with probability generating functions, which help with sums of independent random variables. We then introduce moments, before moving on to moment generating functions.

Among other things, these tools also allow us to calculate probabilities and expected values, and can help us recognise which distribution a random variable has.