added proof "bvn-lincomb"

Author: JoramSoch

P/bvn-lincomb.md

@@ -0,0 +1,80 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-10-25 11:59:37
+
+title: "Linear combination of bivariate normal random variables"
+chapter: "Probability Distributions"
+section: "Multivariate continuous distributions"
+topic: "Multivariate normal distribution"
+theorem: "Linear combination of bivariate normal random variables"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2024
+    title: "Misconceptions about the normal distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2024-10-25"
+    url: "https://en.wikipedia.org/wiki/Misconceptions_about_the_normal_distribution"
+
+proof_id: "P475"
+shortcut: "bvn-lincomb"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ and $Y$ follow a [bivariate normal distribution](/D/bvn):
+
+$$ \label{eq:bvn}
+\left[ \begin{matrix} X \\ Y \end{matrix} \right] \sim
+\mathcal{N}\left( \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right) \; .
+$$
+
+Then, any linear combination of $X$ and $Y$ follows a [univariate normal distribution](/D/norm):
+
+$$ \label{eq:bvn-lincomb}
+Z = a X + b Y \sim
+\mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2ab \sigma_{12} + b^2 \sigma_2^2 \right) \; .
+$$
+
+
+**Proof:** The [linear transformation theorem for the multivariate normal distribution](/P/mvn-ltt) 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
+
+$$ \label{eq:bvn-lincomb-qed}
+\begin{split}
+Z
+&\sim \mathcal{N}\left( \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \left[ \begin{matrix} a & b \end{matrix} \right] \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} a \\ b \end{matrix} \right] \right) \\
+&\sim \mathcal{N}\left( a \mu_1 + b \mu_2, \left[ \begin{matrix} a \sigma_1^2 + b \sigma_{12} & a \sigma_{12} + b \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} a \\ b \end{matrix} \right] \right) \\
+&\sim \mathcal{N}\left( a \mu_1 + b \mu_2, (a^2 \sigma_1^2 + ab \sigma_{12}) + (ab \sigma_{12} + b^2 \sigma_2^2) \right) \\
+&\sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2ab \sigma_{12} + b^2 \sigma_2^2 \right) \; .
+\end{split}
+$$