Cauchy distribution
E239287
continuous probability distribution
heavy-tailed distribution
probability distribution
stable distribution
univariate distribution
The Cauchy distribution is a continuous probability distribution with heavy tails and undefined mean and variance, often used as a classic example of pathological behavior in probability theory and statistics.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cauchy distribution canonical | 3 |
| Breit–Wigner distribution | 1 |
| Lorentz distribution | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171648 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cauchy distribution Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy distribution]
-
A.
Gaussian distribution
The Gaussian distribution, also known as the normal distribution, is a fundamental continuous probability distribution characterized by its symmetric bell-shaped curve and central role in statistics and the natural sciences.
-
B.
F-distribution
The F-distribution is a continuous probability distribution widely used in statistics, especially for comparing variances and conducting analysis of variance (ANOVA) tests.
-
C.
Gumbel
Gumbel is a surname most notably associated with American sportscaster Greg Gumbel.
-
D.
Bernoulli
Bernoulli is the surname of a prominent Swiss family of mathematicians and scientists, including figures such as Jakob, Johann, and Daniel Bernoulli, who made foundational contributions to calculus, probability, and fluid dynamics.
-
E.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cauchy distribution Target entity description: The Cauchy distribution is a continuous probability distribution with heavy tails and undefined mean and variance, often used as a classic example of pathological behavior in probability theory and statistics.
-
A.
Gaussian distribution
The Gaussian distribution, also known as the normal distribution, is a fundamental continuous probability distribution characterized by its symmetric bell-shaped curve and central role in statistics and the natural sciences.
-
B.
F-distribution
The F-distribution is a continuous probability distribution widely used in statistics, especially for comparing variances and conducting analysis of variance (ANOVA) tests.
-
C.
Gumbel
Gumbel is a surname most notably associated with American sportscaster Greg Gumbel.
-
D.
Bernoulli
Bernoulli is the surname of a prominent Swiss family of mathematicians and scientists, including figures such as Jakob, Johann, and Daniel Bernoulli, who made foundational contributions to calculus, probability, and fluid dynamics.
-
E.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
- F. None of above. chosen
Statements (53)
| Predicate | Object |
|---|---|
| instanceOf |
continuous probability distribution
ⓘ
heavy-tailed distribution ⓘ probability distribution ⓘ stable distribution ⓘ univariate distribution ⓘ |
| belongsToFamily | location–scale family ⓘ |
| correspondsToStudentTWithDegreesOfFreedom | 1 ⓘ |
| definedOn | real line ⓘ |
| hasAllMoments | do not exist ⓘ |
| hasAlternativeName |
Cauchy distribution
ⓘ
surface form:
Breit–Wigner distribution
Cauchy distribution ⓘ
surface form:
Lorentz distribution
|
| hasCharacteristicFunction | φ(t) = exp(i x0 t - γ |t|) ⓘ |
| hasConvolutionProperty | sum of independent Cauchy variables is Cauchy ⓘ |
| hasCumulativeDistributionFunction |
F(x) = 1/π arctan(x) + 1/2
ⓘ
F(x; x0, γ) = 1/π arctan((x - x0)/γ) + 1/2 ⓘ |
| hasEntropy | H = ln(4πγ) ⓘ |
| hasHazardFunction | h(x) = f(x) / (1 - F(x)) for x in ℝ ⓘ |
| hasKurtosis | undefined ⓘ |
| hasLocationParameter |
0
ⓘ
x0 ⓘ |
| hasMean | undefined ⓘ |
| hasMedian | x0 ⓘ |
| hasMedianAbsoluteDeviation | γ ⓘ |
| hasMode | x0 ⓘ |
| hasMomentGeneratingFunction | does not exist ⓘ |
| hasProbabilityDensityFunction |
f(x) = 1 / [π (1 + x^2)]
ⓘ
f(x; x0, γ) = 1 / [πγ (1 + ((x - x0)/γ)^2)] ⓘ |
| hasQuantileFunction | Q(p) = x0 + γ tan[π(p - 1/2)] ⓘ |
| hasScaleParameter |
1
ⓘ
γ ⓘ |
| hasSkewness | 0 ⓘ |
| hasStabilityIndex | 1 ⓘ |
| hasStandardForm | standard Cauchy distribution ⓘ |
| hasSupport | (-∞, ∞) ⓘ |
| hasTailBehavior | f(x) ~ 1/(πγ) · 1/x^2 as |x| → ∞ ⓘ |
| hasVariance | undefined ⓘ |
| isHeavyTailed | true ⓘ |
| isLevyAlphaStable | true ⓘ |
| isSpecialCaseOf |
Lévy alpha-stable distribution
ⓘ
Pearson distribution ⓘ
surface form:
Pearson type VII distribution
Student’s t-distribution ⓘ
surface form:
Student's t-distribution
generalized hyperbolic distribution ⓘ q-Gaussian distribution ⓘ |
| isStable | true ⓘ |
| isSymmetricAbout | x0 ⓘ |
| isUsedAs |
example of distribution with undefined mean
ⓘ
example of distribution with undefined variance ⓘ example of pathological behavior in probability theory ⓘ heavy-tailed prior distribution ⓘ |
| isUsedIn |
Bayesian inference
ⓘ
surface form:
Bayesian statistics
robust statistics ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| requiresScaleParameterCondition | γ > 0 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cauchy distribution Description of subject: The Cauchy distribution is a continuous probability distribution with heavy tails and undefined mean and variance, often used as a classic example of pathological behavior in probability theory and statistics.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy
this entity surface form:
Lorentz distribution
this entity surface form:
Breit–Wigner distribution