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

Theorem: Let $X$ be an $n \times 1$ random vector with the moment-generating function $M_X(t)$. Then, the moment-generating function of the linear transformation $Y = A X + b$ is given by

$\label{eq:mgf-ltt} M_Y(t) = \exp \left[ t^\mathrm{T} b \right] \cdot M_X(At)$

where $A$ is an $m \times n$ matrix and $b$ is an $m \times 1$ vector.

Proof: The moment-generating function of a random vector $X$ is

$\label{eq:mfg-vect} M_X(t) = \mathrm{E} \left( \exp \left[ t^\mathrm{T} X \right] \right)$

and therefore the moment-generating function of the random vector $Y$ is given by

$\label{eq:mgf-ltt-qed} \begin{split} M_Y(t) &= \mathrm{E} \left( \exp \left[ t^\mathrm{T} (AX + b) \right] \right) \\ &= \mathrm{E} \left( \exp \left[ t^\mathrm{T} A X \right] \cdot \exp \left[ t^\mathrm{T} b \right] \right) \\ &= \exp \left[ t^\mathrm{T} b \right] \cdot \mathrm{E} \left( \exp \left[ (A t)^\mathrm{T} X \right] \right) \\ &= \exp \left[ t^\mathrm{T} b \right] \cdot M_X(At) \; . \end{split}$
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Metadata: ID: P154 | shortcut: mgf-ltt | author: JoramSoch | date: 2020-08-19, 08:09.