Definition: Cumulant-generating function
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Other probability functions ▷ Cumulant-generating function
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Metadata: ID: D68 | shortcut: cgf | author: JoramSoch | date: 2020-05-31, 23:46.
Definition:
1) The cumulant-generating function of a random variable $X \in \mathbb{R}$ is
\[\label{eq:cgf-var} K_X(t) = \log \mathrm{E} \left[ e^{tX} \right], \quad t \in \mathbb{R} \; .\]2) The cumulant-generating function of a random vector $X \in \mathbb{R}^n$ is
\[\label{eq:cgf-vec} K_X(t) = \log \mathrm{E} \left[ e^{t^\mathrm{T}X} \right], \quad t \in \mathbb{R}^n \; .\]- Wikipedia (2020): "Cumulant"; in: Wikipedia, the free encyclopedia, retrieved on 2020-05-31; URL: https://en.wikipedia.org/wiki/Cumulant#Definition.
Metadata: ID: D68 | shortcut: cgf | author: JoramSoch | date: 2020-05-31, 23:46.