Cumulant generating function

A cumulant generating function is the real-valued function $K_X$ associated to a random variable $X$ whose moment generating function $M_X(t)=\mathbb{E}[e^{tX}]$ is finite on an open interval containing $0$, defined by

for all $t$ where $M_X(t)$ is finite. Derivatives of $K_X$ at $0$ (when they exist) produce the cumulants of $X$; in particular, this links $K_X$ to expectation and variance . If $X$ and $Y$ are independent random variables and both cumulant generating functions exist near $0$, then $K_{X+Y}(t)=K_X(t)+K_Y(t)$.

Examples:

• If $X\sim \mathcal{N}(\mu,\sigma^2)$, then $K_X(t)=\mu t+\tfrac12\sigma^2 t^2$ (finite for all $t\in\mathbb{R}$).
• If $X\sim \mathrm{Bernoulli}(p)$, then $K_X(t)=\log\!\bigl((1-p)+p\,e^{t}\bigr)$.