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If given realization of X and Y, which are i.i.d. Gaussian random variables.

We need to generate random variables C and D which are jointly Gaussian with zero mean and covariance matrix like $ \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$

using X and Y. How to provide a transformation that can be used to obtain C and D from X and Y?

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closed as off-topic by Did, Claude Leibovici, Davide Giraudo, астон вілла олоф мэллбэрг, Namaste Jan 6 '17 at 14:20

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from the covariance matrix $\sigma^2_C=2, \sigma^2_D=2, \sigma_{CD}=\sigma_{DC}=-1$. Hence, $\rho_{CD}=\frac{-1}{2}$. The desired transform is $$\begin{align} C&=\sigma_CX\\ D&=\sigma_D(\rho_{CD}X+\sqrt{1-\rho^2_{CD}}Y) \end{align}$$

It is assumed that $X$ and $Y$ are zero-mean. Otherwise, the same transformation should be used with $\tilde{X}=X-\mu_X$ and $\tilde{Y}=Y-\mu_Y$.

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If $x$ is a vector of i.i.d. Gaussian random variables, and $u=Ax+b$, then $u$ is Gaussian with mean $b$ and covariance matrix $AA^T$.

One way to decompose a given covariance matrix into such a product is the Cholesky decomposition, which msm used for his parametrization $$ \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -\frac12 & 1 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & \frac32 \end{bmatrix} \begin{bmatrix} 1 & -\frac12 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} \sqrt{2} & 0 \\ -\frac{\sqrt{2}}2 & \frac{\sqrt{6}}2 \end{bmatrix} \begin{bmatrix} \sqrt{2} & -\frac{\sqrt{2}}2 \\ 0 & \frac{\sqrt{6}}2 \end{bmatrix} $$

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