Probability Distributions > Degenerate Distribution

## What is a Degenerate Distribution?

A degenerate distribution (sometimes called a*constant distribution*) is a distribution of a degenerate random variable — a constant with probability of 1. In other words, a random variable X has a single possible value.

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 the degenerate distribution 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.

## Use

Degenerate distributions are usually taught in advanced statistics courses like mathematical statistics. They can be defined as special cases of the binomial distribution, normal and geometric distributions among others and are often used in queuing theory where service times or systems interarrival times are constant.

**Reference**:

V. Sundarapandian. Probability, Statistics and Queueing Theory. Dec 1, 2009