Cumulant

In probability theory and statistics, the cumulants κ_n of a probability distribution are given by

$E\left(e^{tX}\right)=\exp \left(\sum _{n=1}^{\infty }\kappa _{n}t^{n}/n!\right)$

where X is any random variable whose probability distribution is the one whose cumulants are taken. In other words, κ_n/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function.

The cumulants of the normal distribution with expected value μ and variance σ² are κ₁ = μ, κ₂ = σ², and κ_n = 0 for n > 2.

All of the cumulants of the Poisson distribution are equal to the expected value.

The first cumulant is shift-equivariant; all of the others are shift-invariant. Somewhat less tersely, if Y = X + c, where c is a constant, and { κ_n } are the cumulants of the probability distribution of X, then the cumulants of the probability distribution of Y are given by κ₁ + c, κ₂, κ₃, etc., i.e., c is added to the first cumulant, but all higher cumulants are unchanged.