cumulant
Sign in to saveIn probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
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- time of discovery or invention
- 1889-00-00
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Article
28 sectionsContents
- Definition
- Alternative definition of the cumulant generating function
- Some basic properties
- First several cumulants as functions of the moments
- Cumulants of some discrete probability distributions
- Cumulants of some continuous probability distributions
- Some properties of the cumulant generating function
- Further properties of cumulants
- A negative result
- Cumulants and moments
- Cumulants and set-partitions
- Cumulants and combinatorics
- Joint cumulants
- Repeated random variables and relation between the coefficients ''κ''<sub>''k''<sub>1</sub>, ..., ''k''<sub>''n''</sub></sub>
- Relation with mixed moments
- Further properties
- Conditional cumulants and the law of total cumulance
- Conditional cumulants and the conditional expectation
- Relation to statistical physics
- History
- Cumulants in generalized settings
- Formal cumulants
- Bell numbers
- Cumulants of a polynomial sequence of binomial type
- Free cumulants
- See also
- References
- External links
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa.
The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the th-order cumulant of their sum is equal to the sum of their th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.