Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are given by
where X is any random variable whose probability distribution is the one whose cumulants are taken. In other words, κn/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 κn = 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 { κn } 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.