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Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...

Let $X \sim \mathcal{N}\left( {{\mu _x},\sigma _x^2} \right)$ and $Y \sim \mathcal{N}\left( {{\mu _y},\sigma _y^2} \right)$ be two univariate and independent Gaussian/normal random variables and let $...