added proof "norm-margjoint"

Author: JoramSoch

P/norm-margjoint.md

@@ -0,0 +1,114 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-10-11 11:52:29
+
+title: "Marginally normal does not imply jointly normal"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Normal distribution"
+theorem: "Marginally normal does not imply jointly normal"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2024
+    title: "Misconceptions about the normal distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2024-10-11"
+    url: "https://en.wikipedia.org/wiki/Misconceptions_about_the_normal_distribution#A_symmetric_example"
+
+proof_id: "P474"
+shortcut: "norm-margjoint"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Consider two [random variables](/D/rvar) $X$ and $Y$. If the [marginal distribution](/D/dist-marg) of each of them is a [normal distribution](/D/norm), then the [joint distribution](/D/dist-joint) $X$ and $Y$ is not necessarily a [(multivariate) normal distribution](/D/mvn).
+
+
+**Proof:** Consider [the example used to show that normally distributed and uncorrelated does not imply independent](/P/norm-corrind). This is characterized by the [random variables](/D/rvar)
+
+$$ \label{eq:V-W}
+\begin{split}
+V &\sim \mathrm{Bern}\left( \frac{1}{2} \right) \\
+W &= 2V-1 \; .
+\end{split}
+$$
+
+and
+
+$$ \label{eq:X-Y}
+\begin{split}
+X &\sim \mathcal{N}(0,1) \\
+Y &= WX \; .
+\end{split}
+$$
+
+Under these conditions, [it can be shown that](/P/norm-corrind)
+
+$$ \label{eq:X-Y-dist}
+X \sim \mathcal{N}(0,1)
+\quad \text{and} \quad
+Y \sim \mathcal{N}(0,1) \; .
+$$
+
+The [linear transformation theorem for the multivariate normal distribution](/P/mvn-ltt)
+
+$$ \label{eq:mvn-ltt}
+x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T})
+$$
+
+implies that, for [bivariate normal random variables](/D/bvn) $X_1$ and $X_2$,
+
+$$ \label{eq:bvn}
+\left[ \begin{matrix} X_1 \\ X_2 \end{matrix} \right] \sim
+\mathcal{N}\left(
+\left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right],
+\left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right]
+\right) \; .
+$$
+
+any linear combination of $X_1$ and $X_2$ with non-zero coefficients
+
+$$ \label{eq:bvn-Z}
+Z = a X_1 + b X_2, \; a \neq 0, \; b \neq 0
+$$
+
+[follows a univariate normal distribution](/P/bvn-lincomb):
+
+$$ \label{eq:bvn-lincomb}
+Z \sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2 a b \sigma_{12} + b^2 + \sigma_2^2 \right) \; .
+$$
+
+Consider the sum of $X$ and $Y$ defined by \eqref{eq:X-Y}:
+
+$$ \label{eq:Z}
+Z = X + Y = a X + b Y
+\quad \text{with} \quad
+a = b = 1 \; .
+$$
+
+If $X$ and $Y$ were [bivariate normally distributed](/D/bvn), then this sum should be [univariate normally distributed](/D/norm). However, this sum cannot be normally distributed, since
+
+$$ \label{eq:Z-dist}
+\mathrm{Pr}(X + Y = 0)  = \frac{1}{2}
+\quad \text{and} \quad
+\mathrm{Pr}(X + Y = 2X) = \frac{1}{2} \; ,
+$$
+
+because
+
+$$ \label{eq:Y-dist}
+Y = \left\{
+\begin{array}{rl}
+ X \; , & \text{with probability} \; 1/2 \\
+-X \; , & \text{with probability} \; 1/2
+\end{array}
+\right. \; .
+$$
+
+Thus, $X$ and $Y$ are not following a [bivariate normal distribution](/D/bvn). Therefore, $X$ and $Y$ defined by \eqref{eq:X-Y} and \eqref{eq:V-W} constitute an example for two [random variables](/D/rvar) that are [marginally normal](/D/norm), but not [jointly normal](/D/mvn).