# Degenerate Distribution: Simple Definition & Examples

Probability Distributions > Degenerate Distribution

## What is a Degenerate Distribution?

A degenerate distribution (sometimes called a constant distribution) is a probability distribution of a degenerate random variable — a constant with probability of 1. In other words, a random variable X has a single possible value, with all of the mass concentrated at a single point.
A few examples:
• A weighted die (or one that has a number 6 on all faces) always lands on the number six, so the probability of a six (P(6)) is 1.
• A coin is double-sided with two heads (thousands of these “magician’s coins” exist, but there are also real ones. See: 1859 Double-Headed Indian Head Cent for an example).
• (Calculus): A random variable X that is distributed as the derivative of k when k=1. As k=1, the distribution can only have a value of 0 (because the derivative of any constant is 0).
As there is no spread of variables around the mean, the variance for is zero (Var(X) = 0).

## More Formal Definitions

The formal definition of a degenerate random variable is that it’s a distribution assigning all of the probability to a single point:
A random variable, X, is degenerate if, for some a constant, c, P(X = c) = 1
If a random variable does not meet the above definition, then it is non-degenerate. A degenerate distribution has a single parameter, c, where -∞ < c < ∞. The formal definition is:
for some a constant, x, F(x) =
• 0, for x < c,
• 1, for x ≥ c.
A non-degenerate distribution is one that doesn’t meet this definition. In general, there are two types of degenerate distributions: discrete and continuous:
• Discrete: This 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.
• Continuous: this happens 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.

## Uses for a degenerate distribution

Degenerate distributions are usually taught in advanced statistics courses like mathematical statistics. They can be defined as special cases of the binomial distribution, normal distribution and geometric distributions among others and are often used in queuing theory where service times or systems interarrival times are constant. These 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: one that is degenerate with mass at zero and a Poisson distribution [1]. Other uses include setting bounds for probability distributions [2].

## References

1. Perraillon, M. Finite Mixture Models
2. Barlow, R. & Marshall, A. BOUNDS ON INTERVAL PROBABILITIES FOR RESTRICTED FAMILIES OF DISTRIBUTIONS.