For bivariate validation, 160,000 conditions were tested, consisting of 20 values each for $\mu_x$ and $\mu_y$ evenly distributed between 0 and 10, 20 values for $\sigma_x$ evenly distributed between 0.25 and 10, and 20 values for the ratio $\frac{\sigma_y}{\sigma_x}$ between 1 and 10. For each condition, 1,000,000 data were drawn from a bivariate normal distribution $X \sim \mathcal{N}_2(\mu,\Sigma)$. The mean and variance of the Euclidean distance error between generated data and the origin were calculated, and the difference between the Monte Carlo solutions and those obtained from the following equations: $$E[\sqrt{X_1^2+X_2^2}] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\sqrt{x_1^2+x_2^2} f(x|\mu,\Sigma)dx_2dx_1$$ $$Var(\sqrt{X_1^2+X_2^2}) = E[X_1^2+X_2^2] - E[\sqrt{X_1^2+X_2^2}]^2$$ $$E[X_1^2+X_2^2] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x_1^2+x_2^2) f(x|\mu,\Sigma)dx_2dx_1$$ The difference between simulated and calculated values were then normalized by the mean error of the simulated data.