Cumulant

The cumulant $\kappa_r$ .

Cumulant(dist, r)exact for

r=1

(=Mean) and

r=2

(=Variance) for every distribution — the only two orders with a formula that doesn't need higher raw moments this generic layer doesn't have.

r=0

and

r\geq 3

stay unevaluated EXCEPT PoissonDistribution, whose every cumulant equals its rate.

Typed signatures

Type	Package	Applies
`(distribution, integer) -> real`	`statistics`	always

Domain: Statistics

Details

Generic over every distribution this package's Mean/Variance answer, old or new.
PoissonDistribution( $\lambda$ ): $\log E[e^{tX}] = \lambda(e^t-1)$ , whose Taylor coefficients are all $\lambda$ — so $Cumulant(\lambda, r) = \lambda$ at every order $r\geq1$ , not just $r\leq2$ .

Examples

Basic #

\mathrm{Cumulant}(\mathrm{ChiSquareDistribution}(3), 1)

Out

3

\mathrm{Cumulant}(\mathrm{ChiSquareDistribution}(3), 2)

Out

6

\mathrm{Cumulant}(\operatorname{PoissonDistribution}(3), 5)

Out

3

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Examples	3