Zero-modified counting distributions

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See also: Distributions introduction, Discrete distributions introduction, Distributions in ModelRisk

A counting distribution is a discrete distribution with only non-negative integers in its domain. We typically use a counting distribution to model the number of occurrences of a certain event, for example "number of car accidents in a year". The Poisson distribution is a counting distribution for example.

However, it can occur that "zero events occurring" is not properly modeled by the counting distribution. A solution for this is to use a zero-modified distribution, which alters the probability of occurrence of zero. There are basically two ways of modifying the counting distribution at zero:

When we assign an extra probability to zero compared to the original distribution we have a zero-inflated model. The best-known example of a zero-inflated model is the zero-inflated-Poisson or ZIP model.

If we remove the probability of zero occurring (i.e. P(0)=0) then we speak of a zero-truncated model:

Zero-inflated (discrete) distributions are distributions with an extra probability assigned to zero as an outcome. Suppose we have a distribution with probability mass function f(x), and let . Then we define the probability mass of the Zero-Inflated (ZI) distribution as:

The graph below shows the difference between a ZI, ZT and unmodified Poisson(2) distribution:

Properties

Most properties (e.g. the moments) of a zero-modified distribution follow easily from its unmodified counterpart. Things like the raw moments, moments-generating function, etc... all have a fairly simple form. However, when calculating the central moments, the formulae quickly become complicated, which is why you will often see 'complicated' occur in the ZI and ZT distribution equations (listed on the right).