Degenerate distribution

A probability distribution with all mass concentrated at a single point

Degenerate distribution

A degenerate distribution is a probability distribution in which all the probability mass is concentrated at a single point. This means that the random variable associated with this distribution is almost surely equal to a constant value. In other words, the outcome is deterministic.

Illustration of a degenerate distribution

Definition

Formally, a degenerate distribution is defined for a random variable \(X\) such that:

\[ P(X = x_0) = 1 \]

for some constant \(x_0\). This implies that for any other value \(x \neq x_0\), the probability is zero:

\[ P(X = x) = 0 \]

Properties

• Mean: The mean of a degenerate distribution is the constant value \(x_0\) at which all the probability mass is concentrated.
• Variance: The variance of a degenerate distribution is zero, as there is no variability in the outcomes.
• Support: The support of a degenerate distribution is the singleton set \(\{x_0\}\).

Examples

• A coin that always lands on heads is an example of a degenerate distribution, where the outcome is always "heads".
• A die that always rolls a six is another example, with the outcome always being the number six.

Applications

Degenerate distributions are often used in probability theory and statistics to model deterministic processes. They are also used in stochastic processes as a limiting case where randomness is absent.

Related concepts

• Dirac measure: A measure that assigns all mass to a single point, similar to a degenerate distribution.
• Deterministic system: A system in which no randomness is involved in the development of future states.

Related pages

• Probability distribution
• Random variable
• Variance
• Mean