Proof: Marginally normal does not imply jointly normal
Theorem: Consider two random variables $X$ and $Y$. If the marginal distribution of each of them is a normal distribution, then the joint distribution $X$ and $Y$ is not necessarily a (multivariate) normal distribution.
Proof: Consider the example used to show that normally distributed and uncorrelated does not imply independent. This is characterized by the random variables
\[\label{eq:V-W} \begin{split} V &\sim \mathrm{Bern}\left( \frac{1}{2} \right) \\ W &= 2V-1 \; . \end{split}\]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
\[\label{eq:X-Y-dist} X \sim \mathcal{N}(0,1) \quad \text{and} \quad Y \sim \mathcal{N}(0,1) \; .\]The linear transformation theorem for the multivariate normal distribution
\[\label{eq:mvn-ltt} x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T})\]implies that, for bivariate normal random variables $X_1$ 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:
\[\label{eq:bvn-lincomb} Z \sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2 a b \sigma_{12} + b^2 \sigma_2^2 \right) \; .\]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, then this sum should be univariate normally distributed. 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. Therefore, $X$ and $Y$ defined by \eqref{eq:X-Y} and \eqref{eq:V-W} constitute an example for two random variables that are marginally normal, but not jointly normal.
- Wikipedia (2024): "Misconceptions about the normal distribution"; in: Wikipedia, the free encyclopedia, retrieved on 2024-10-11; URL: https://en.wikipedia.org/wiki/Misconceptions_about_the_normal_distribution#A_symmetric_example.
Metadata: ID: P474 | shortcut: norm-margjoint | author: JoramSoch | date: 2024-10-11, 11:52.