# Degenerate Distribution

< 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 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.

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