Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability functions ▷ Moment-generating function

Definition:

1) The moment-generating function of a random variable $X \in \mathbb{R}$ is

\[\label{eq:mgf-var} M_X(t) = \mathrm{E} \left[ e^{tX} \right], \quad t \in \mathbb{R} \; .\]

2) The moment-generating function of a random vector $X \in \mathbb{R}^n$ is

\[\label{eq:mgf-vec} M_X(t) = \mathrm{E} \left[ e^{t^\mathrm{T}X} \right], \quad t \in \mathbb{R}^n \; .\]
 
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Metadata: ID: D2 | shortcut: mgf | author: JoramSoch | date: 2020-01-22, 10:58.