Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Linear transformation

Theorem: Let $x$ follow a multivariate normal distribution:

\[\label{eq:mvn} x \sim \mathcal{N}(\mu, \Sigma) \; .\]

Then, any linear transformation of $x$ is also multivariate normally distributed:

\[\label{eq:mvn-lt} y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T}) \; .\]

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

\[\label{eq:vect-mgf} M_x(t) = \mathbb{E} \left( \exp \left[ t^\mathrm{T} x \right] \right)\]

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

\[\label{eq:y-mgf-s1} \begin{split} M_y(t) &\overset{\eqref{eq:mvn-lt}}{=} \mathbb{E} \left( \exp \left[ t^\mathrm{T} (Ax + b) \right] \right) \\ &= \mathbb{E} \left( \exp \left[ t^\mathrm{T} A x \right] \cdot \exp \left[ t^\mathrm{T} b \right] \right) \\ &= \exp \left[ t^\mathrm{T} b \right] \cdot \mathbb{E} \left( \exp \left[ t^\mathrm{T} A x \right] \right) \\ &\overset{\eqref{eq:vect-mgf}}{=} \exp \left[ t^\mathrm{T} b \right] \cdot M_x(A^\mathrm{T} t) \; . \end{split}\]

The moment-generating function of the multivariate normal distribution is

\[\label{eq:mvn-mgf} M_x(t) = \exp \left[ t^\mathrm{T} \mu + \frac{1}{2} t^\mathrm{T} \Sigma t \right]\]

and therefore the moment-generating function of the random vector $y$ becomes

\[\label{eq:y-mgf-s2} \begin{split} M_y(t) &\overset{\eqref{eq:y-mgf-s1}}{=} \exp \left[ t^\mathrm{T} b \right] \cdot M_x(A^\mathrm{T} t) \\ &\overset{\eqref{eq:mvn-mgf}}{=} \exp \left[ t^\mathrm{T} b \right] \cdot \exp \left[ t^\mathrm{T} A \mu + \frac{1}{2} t^\mathrm{T} A \Sigma A^\mathrm{T} t \right] \\ &= \exp \left[ t^\mathrm{T} \left( A \mu + b \right) + \frac{1}{2} t^\mathrm{T} A \Sigma A^\mathrm{T} t \right] \; . \end{split}\]

Because moment-generating function and probability density function of a random variable are equivalent, this demonstrates that $y$ is following a multivariate normal distribution with mean $A \mu + b$ and covariance $A \Sigma A^\mathrm{T}$.

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Metadata: ID: P1 | shortcut: mvn-ltt | author: JoramSoch | date: 2019-08-27, 12:14.