# From Bivariate to Multivariate Normal Distribution: An additive representation

Let $X=Z_1$ and $Y=\rho Z_1 + \sqrt{1-\rho^2}Z_2$ where $Z_1$ and $Z_2$ follow two independent standard normal distribution, $\mathbb{N}(0,1)$. This means that $\begin{pmatrix} X \\ Y \end{pmatrix}=\mathbb{N}\left(\begin{pmatrix} 0 \\ 0 \end{pmatrix},\begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}\right)$
Say it differently, X and Y follow the standard normal distribution with correlation $\rho$.
Now, let's move the multivariate case. How can $Y$ be written if we suppose that $X=[X_1,\ldots,X_n]$ is now a vector of independent i.i.d standard normals:
$\begin{pmatrix} X \\ Y \end{pmatrix}=\mathbb{N}\left(\begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix},\begin{pmatrix} 1 & 0 & \ldots & 0 & \rho_1 \\ 0 & 1 & \ldots & 0 & \rho_2 \\ \vdots & 0 & \ldots & 0 & \vdots \\ 0 & 0 & \ldots & 1 & \rho_{n} \\ \rho_1 & \rho_2 & \ldots & \rho_{n} & 1 \\ \end{pmatrix}\right)$
• The variance matrix in the two-variable case is not correctly presented. If you would fix it, the parallels between that and the $n+1$-variable case should be so clear that the answer to your question ought to be apparent. – whuber Jan 2 '18 at 14:54