9.4 Integer Distribution Families

Mathematically, integer distributions are commonly defined in one of two ways: over extended reals, or over extended integers. The most common definitions use the extended reals, so the following distribution object constructors return objects of type Real-Dist.

(Another reason is that the extended integers correspond with the type (U Integer +inf.0 -inf.0). Values of this type have little support in Racket’s library.)

This leaves us with a quandary and two design decisions users should be aware of. The quandary is that, when an integer distribution is defined over the reals, it has a cdf, but no well-defined pdf: the pdf would be zero except at integer points, where it would be undefined.

Unfortunately, an integer distribution without a pdf is nearly useless. In measure-theory parlance, the pdfs are defined with respect to counting measure, while the cdfs are defined with respect to Lebesgue measure. So the pdfs of these integer distributions are pdfs defined over integers, while their cdfs are defined over reals.

Integer distributions defined over the extended integers are not out of the question, and may show up in future versions of math/distributions if there is a clear need.

9.4.1 Bernoulli Distributions

(bernoulli-dist prob) is equivalent to (binomial-dist 1 prob), but operations on it are faster.

9.4.2 Binomial Distributions

9.4.3 Geometric Distributions

9.4.4 Poisson Distributions