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For univariate validation, 10,000 conditions were tested, consisting of 100 values for $\mu$ evenly distributed between 0 and 10, and 100 values for $\sigma$ evenly distributed between 0.25 and 10. For each condition, 1,000,000 data were drawn from a univariate normal distribution $X \sim \mathcal{N}(μ,σ^2)$. The mean and variance of the error between generated data and the origin were calculated, and the difference between the Monte Carlo solution and those obtained from the following equations: $$\mu_f = \sigma \sqrt{\frac{2}{\pi}} e^{\frac{-\mu^2}{2\sigma^2}} + \mu(1-2\Phi(\frac{-\mu}{\sigma}))$$ $$\sigma_f^2 = \mu^2 + \sigma^2 - \mu_f^2$$ The difference between simulated and calculated values were then normalized by the mean error of the simulated data.
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