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

Theorem: Let $X$ and $Y$ follow a bivariate normal distribution:

\[\label{eq:bvn} \left[ \begin{matrix} X \\ Y \end{matrix} \right] \sim \mathcal{N}\left( \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right) \; .\]

Then, any linear combination of $X$ and $Y$ follows a univariate normal distribution:

\[\label{eq:bvn-lincomb} Z = a X + b Y \sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2ab \sigma_{12} + b^2 \sigma_2^2 \right) \; .\]

Proof: The linear transformation theorem for the multivariate normal distribution states that

\[\label{eq:mvn-ltt} X \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad Y = AX + c \sim \mathcal{N}(A\mu + c, A \Sigma A^\mathrm{T})\]

where $X \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$ and $c \in \mathbb{R}^n$. In the present case, we have

\[\label{eq:X-A-a} X \in \mathbb{R}^2 \quad \text{and} \quad A = \left[ \begin{matrix} a & b \end{matrix} \right] \in \mathbb{R}^{1 \times 2} \quad \text{and} \quad c = 0 \in \mathbb{R} \; ,\]

such that

\[\label{eq:Z-X-Y} Z = A \left[ \begin{matrix} X \\ Y \end{matrix} \right] + c = \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} X \\ Y \end{matrix} \right] + 0 = a X + b Y \; .\]

Combining \eqref{eq:mvn-ltt}, \eqref{eq:bvn} and \eqref{eq:Z-X-Y}, it follows that

\[\label{eq:bvn-lincomb-qed} \begin{split} Z &\sim \mathcal{N}\left( \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} a \\ b \end{matrix} \right] \right) \\ &\sim \mathcal{N}\left( a \mu_1 + b \mu_2, \left[ \begin{matrix} a \sigma_1^2 + b \sigma_{12} & a \sigma_{12} + b \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} a \\ b \end{matrix} \right] \right) \\ &\sim \mathcal{N}\left( a \mu_1 + b \mu_2, (a^2 \sigma_1^2 + ab \sigma_{12}) + (ab \sigma_{12} + b^2 \sigma_2^2) \right) \\ &\sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2ab \sigma_{12} + b^2 \sigma_2^2 \right) \; . \end{split}\]
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Metadata: ID: P475 | shortcut: bvn-lincomb | author: JoramSoch | date: 2024-10-25, 11:59.