Degenerate distribution

Degenerate distribution is a concept in probability theory and statistics that describes a probability distribution of a random variable that only takes a single value. This type of distribution is considered "degenerate" because it does not exhibit the variability typically associated with a probability distribution. In essence, a degenerate distribution is a distribution where the probability mass function (PMF) for a discrete variable, or the probability density function (PDF) for a continuous variable, assigns a probability of one to a single value and zero to all other values.

Definition

For a discrete random variable X, the PMF of a degenerate distribution at a value k is defined as:

P(X = k) = 1

And for all other values x ≠ k:

P(X = x) = 0

Similarly, for a continuous random variable X, the PDF would be represented using the Dirac delta function to indicate that all the probability mass is concentrated at a single point k.

Properties

Degenerate distributions have several key properties that distinguish them from other probability distributions:

• Variance: The variance of a degenerate distribution is zero, as there is no variability in the values of the random variable.
• Expectation: The expected value (or mean) of a degenerate distribution is equal to the single value k that the distribution takes.
• Lack of randomness: Since the outcome of a degenerate distribution is always known, it lacks the randomness associated with other distributions.

Applications

Degenerate distributions are often used in theoretical work to simplify calculations or to represent certain deterministic outcomes within a probabilistic framework. For example, they can be used in compound distributions where one of the components is deterministic, or in Bayesian statistics as prior distributions when there is certainty about the value of a parameter.

Examples

A simple example of a degenerate distribution is the distribution of a fair die that has been modified so that it always lands on the number 4. In this case, the probability of rolling a 4 is 1, and the probability of rolling any other number is 0.

See Also

• Probability distribution
• Random variable
• Variance
• Expected value
• Dirac delta function