Cumulant

A cumulant (of order $n$) of a random variable $X$ is the number

provided the cumulant generating function $K_X$ exists in a neighborhood of $0$ and is $n$ times differentiable at $0$. The first two cumulants are $\kappa_1(X)=\mathbb{E}[X]$ and $\kappa_2(X)=\mathrm{Var}(X)$, linking cumulants to expectation and variance . Cumulants are additive over sums of independent random variables (when defined), which makes them useful for studying distributional limits and approximations.

Examples:

• If $X\sim \mathcal{N}(\mu,\sigma^2)$, then $\kappa_1(X)=\mu$, $\kappa_2(X)=\sigma^2$, and $\kappa_n(X)=0$ for all $n\ge 3$.
• If $X\sim \mathrm{Bernoulli}(p)$, then $\kappa_1(X)=p$, $\kappa_2(X)=p(1-p)$, and $\kappa_3(X)=p(1-p)(1-2p)$.