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Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$),

how to find the best Gaussian/normal approximation without drawing samples from the density function $f(x)$ via rejection method and then computing the mean and variance?

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Best approximation in what sense? – Robert Israel Feb 17 '13 at 6:22
@RobertIsrael minimize KL-divergence – Hugo Feb 17 '13 at 7:57

There's no need for rejection – you can estimate the mean and variance by numerical quadrature. That only requires some function evaluations; if even that is too expensive, I don't see what you could possibly do, since you'd essentially have a mystery function in a black box that you can't know anything about.

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