< List of probability distributions > *Degenerate Distribution*

## What is a Degenerate Distribution?

A **degenerate distribution** (sometimes called a* constant distribution*) is a discrete probability distribution where a random variable X has only one possible value. In other words, all of the mass is concentrated at a single point.

We can write this more formally as follows:

*A random variable, X, is degenerate if, for some a constant, c, P(X = c) = 1*.

For example, if a weighted die that always lands on 6, the result is a degenerate probability distribution because the distribution has just one possible outcome — (P(6)) = 1.

Other examples:

- A coin with heads on both sides always lands on heads: P(heads) = 1.
- A random variable
*X*is distributed as the derivative of*k*when*k*= 1. The derivative of a constant is 0, so the distribution can only have a value of 0: P(0) = 1.

Degenerate distributions sometimes pop up in finite mixture models. For example, the zero-inflated Poisson model (ZIP) assumes that the observed data comes from a mixture of two distributions: a degenerate distribution with mass at zero and a Poisson distribution [1]. Other uses include setting bounds for probability distributions [2].

## Degenerate distribution properties

A degenerate distribution has one parameter, *c*. For any variable, *x*, the probability mass function (PMF) is

- F(
*x*) = 0, for x <*c*, - F(
*x*) = 1, for*x*≥*c*.

In general, there are two types of degenerate distributions: discrete and continuous:

- A
**discrete degenerate distribution**occurs when the random variable X takes on values from a finite set. This means that the probability function for the random variable X can take on only certain values or ranges of values. - A
**continuous degenerate distribution**occurs when the random variable X takes on values from an infinite set. This means that the probability function for the random variable X can take on any real number between two given numbers.

## References

[1] Perraillon, M. Finite Mixture Models

[2] Barlow, R. & Marshall, A. BOUNDS ON INTERVAL PROBABILITIES FOR RESTRICTED FAMILIES OF DISTRIBUTIONS.