Bivariate Gaussian

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The bivariate Gaussian can be written as:

f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp \left( -\frac{1}{2 (1-\rho^2)} \left[ \frac{\left(x - \mu_x\right)^2}{\sigma_x^2} + \frac{\left(y - \mu_y\right)^2}{\sigma_y^2} - \frac{2 \rho \left(x - \mu_x\right) \left(y - \mu_y\right)}{ (\sigma_x \sigma_y)} \right] \right)

where ρ is the correlation between X and Y. In other words

\Sigma = \begin{bmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{bmatrix}.
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