Degenerate distribution: Difference between revisions

Latest revision as of 11:20, 15 February 2025

A probability distribution with all mass concentrated at a single point

Degenerate distribution[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

@@ Line 1: / Line 1: @@
-'''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.
+{{short description|A probability distribution with all mass concentrated at a single point}}
-==Definition==
+== Degenerate distribution ==
-For a discrete random variable ''X'', the PMF of a degenerate distribution at a value ''k'' is defined as:
+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.
-:P(''X'' = ''k'') = 1
+[[File:Degenerate.svg|thumb|right|Illustration of a degenerate distribution]]
-And for all other values ''x'' ≠ ''k'':
+== Definition ==
+Formally, a degenerate distribution is defined for a random variable \(X\) such that:
-:P(''X'' = ''x'') = 0
+\[
+P(X = x_0) = 1
+\]
-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''.
+for some constant \(x_0\). This implies that for any other value \(x \neq x_0\), the probability is zero:
-==Properties==
+\[
-Degenerate distributions have several key properties that distinguish them from other probability distributions:
+P(X = x) = 0
+\]
-* '''Variance''': The variance of a degenerate distribution is zero, as there is no variability in the values of the random variable.
+== Properties ==
-* '''Expectation''': The expected value (or mean) of a degenerate distribution is equal to the single value ''k'' that the distribution takes.
+* '''Mean''': The mean of a degenerate distribution is the constant value \(x_0\) at which all the probability mass is concentrated.
-* '''Lack of randomness''': Since the outcome of a degenerate distribution is always known, it lacks the randomness associated with other distributions.
+* '''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 \(\{x_0\}\).
-==Applications==
+== Examples ==
-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.
+* 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.
-==Examples==
+== Applications ==
-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.
+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.
-==See Also==
+== 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]]
-* [[Expected value]]
+* [[Mean]]
-* [[Dirac delta function]]
 [[Category:Probability distributions]]
-[[Category:Statistical theory]]
-{{Probability-stub}}