Cumulant generating function - Imported Knowlpedia Content

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

K_X(t) \;=\; \log M_X(t),

for all $t$ where $M_X(t)$ is finite.

Remarks

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 $K_X(t)=\log\!\bigl((1-p)+p\,e^{t}\bigr)$ .