Zero-Modified Distribution

< List of probability distributions

A zero-modified distribution is a probability distribution that starts at zero. More specifically, it is a distribution modified to put extra probability mass at zero.

Any distribution of this type has the “zero-modified” prefix; they are commonly used to model data that has a high occurrence of zero counts.

Probability mass function for a zero-modified binomial distribution [1].

A similar distribution is the zero-truncated distribution. the difference is that the the probability of zero occurring in a zero-truncated distribution is 0:

  • Zero-modified distribution: p0 > 0
  • Zero-truncated distribution p0 = 0.

All zero-truncated distributions are members of the zero-modified family.

Zero-modified counting distribution

Counting distributions, such as the Poisson distribution, are discrete distributions that have a domain of only non-negative integers {1, 2, 3, …). These types of distribution are often used to model the number of event occurrences such as the number of hurricanes in a calendar year.

Counting distributions cannot be used to model events that include zero. For example, if zero hurricanes happen in a year, the Poisson distribution — without modification — is not suitable. That’s because probabilities are always greater than 0 — never equal to zero. Thus, one option is to modify the distribution to include zero in the domain — giving a mixture model of a standard Poisson distribution and a degenerate distribution at zero.

For example, the probability mass function (PMF) of a zero-modified Poisson distribution is p(0) = p0 and:

PMF for the zero-modified Poisson distribution [2].

Where f(x), the Poisson distribution PMF, is modified by a constant. The constant assigns a large value to p(0) and modifies the other values of p(x). In actuarial science, this type of distribution is called class (a, b, 1) or the zero-modified distribution of the base (a,b,0) distribution [3].


[1] R-forge distributions Core Team. Handbook on Probability Distributions.

[2] R Documentation. The Zero-Modified Poisson Distribution. Retrieved March 5, 2023 from:

[3] Discrete Distributions Chapter 6.