|
| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Joram Soch" |
| 6 | +affiliation: "BCCN Berlin" |
| 7 | +e_mail: "joram.soch@bccn-berlin.de" |
| 8 | +date: 2024-10-11 11:52:29 |
| 9 | + |
| 10 | +title: "Marginally normal does not imply jointly normal" |
| 11 | +chapter: "Probability Distributions" |
| 12 | +section: "Univariate continuous distributions" |
| 13 | +topic: "Normal distribution" |
| 14 | +theorem: "Marginally normal does not imply jointly normal" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Wikipedia" |
| 18 | + year: 2024 |
| 19 | + title: "Misconceptions about the normal distribution" |
| 20 | + in: "Wikipedia, the free encyclopedia" |
| 21 | + pages: "retrieved on 2024-10-11" |
| 22 | + url: "https://en.wikipedia.org/wiki/Misconceptions_about_the_normal_distribution#A_symmetric_example" |
| 23 | + |
| 24 | +proof_id: "P474" |
| 25 | +shortcut: "norm-margjoint" |
| 26 | +username: "JoramSoch" |
| 27 | +--- |
| 28 | + |
| 29 | + |
| 30 | +**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). |
| 31 | + |
| 32 | + |
| 33 | +**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) |
| 34 | + |
| 35 | +$$ \label{eq:V-W} |
| 36 | +\begin{split} |
| 37 | +V &\sim \mathrm{Bern}\left( \frac{1}{2} \right) \\ |
| 38 | +W &= 2V-1 \; . |
| 39 | +\end{split} |
| 40 | +$$ |
| 41 | + |
| 42 | +and |
| 43 | + |
| 44 | +$$ \label{eq:X-Y} |
| 45 | +\begin{split} |
| 46 | +X &\sim \mathcal{N}(0,1) \\ |
| 47 | +Y &= WX \; . |
| 48 | +\end{split} |
| 49 | +$$ |
| 50 | + |
| 51 | +Under these conditions, [it can be shown that](/P/norm-corrind) |
| 52 | + |
| 53 | +$$ \label{eq:X-Y-dist} |
| 54 | +X \sim \mathcal{N}(0,1) |
| 55 | +\quad \text{and} \quad |
| 56 | +Y \sim \mathcal{N}(0,1) \; . |
| 57 | +$$ |
| 58 | + |
| 59 | +The [linear transformation theorem for the multivariate normal distribution](/P/mvn-ltt) |
| 60 | + |
| 61 | +$$ \label{eq:mvn-ltt} |
| 62 | +x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T}) |
| 63 | +$$ |
| 64 | + |
| 65 | +implies that, for [bivariate normal random variables](/D/bvn) $X_1$ and $X_2$, |
| 66 | + |
| 67 | +$$ \label{eq:bvn} |
| 68 | +\left[ \begin{matrix} X_1 \\ X_2 \end{matrix} \right] \sim |
| 69 | +\mathcal{N}\left( |
| 70 | +\left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], |
| 71 | +\left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] |
| 72 | +\right) \; . |
| 73 | +$$ |
| 74 | + |
| 75 | +any linear combination of $X_1$ and $X_2$ with non-zero coefficients |
| 76 | + |
| 77 | +$$ \label{eq:bvn-Z} |
| 78 | +Z = a X_1 + b X_2, \; a \neq 0, \; b \neq 0 |
| 79 | +$$ |
| 80 | + |
| 81 | +[follows a univariate normal distribution](/P/bvn-lincomb): |
| 82 | + |
| 83 | +$$ \label{eq:bvn-lincomb} |
| 84 | +Z \sim \mathcal{N}\left( a \mu_1 + b \mu_2, a^2 \sigma_1^2 + 2 a b \sigma_{12} + b^2 + \sigma_2^2 \right) \; . |
| 85 | +$$ |
| 86 | + |
| 87 | +Consider the sum of $X$ and $Y$ defined by \eqref{eq:X-Y}: |
| 88 | + |
| 89 | +$$ \label{eq:Z} |
| 90 | +Z = X + Y = a X + b Y |
| 91 | +\quad \text{with} \quad |
| 92 | +a = b = 1 \; . |
| 93 | +$$ |
| 94 | + |
| 95 | +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 |
| 96 | + |
| 97 | +$$ \label{eq:Z-dist} |
| 98 | +\mathrm{Pr}(X + Y = 0) = \frac{1}{2} |
| 99 | +\quad \text{and} \quad |
| 100 | +\mathrm{Pr}(X + Y = 2X) = \frac{1}{2} \; , |
| 101 | +$$ |
| 102 | + |
| 103 | +because |
| 104 | + |
| 105 | +$$ \label{eq:Y-dist} |
| 106 | +Y = \left\{ |
| 107 | +\begin{array}{rl} |
| 108 | + X \; , & \text{with probability} \; 1/2 \\ |
| 109 | +-X \; , & \text{with probability} \; 1/2 |
| 110 | +\end{array} |
| 111 | +\right. \; . |
| 112 | +$$ |
| 113 | + |
| 114 | +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). |
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