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I need to evaluate the probability $P =c\times \textrm{Pr}\{X < a\}$ where $c$ is a constant and the PDF of $X$ is not available. Then it is natural I resort to Monte Carlo simulation to evaluate $\textrm{Pr}\{X < a\}$.

The problem is that $\textrm{Pr}\{X < a\}$ is small, say $10^{-10}$, while $c$ is quite large, about $10^{4}$, hence $P$ is just about $10^{-6}$. It is painful to simulate $10^{+10}$ samples just to get something at order $10^{-6}$.

Any trick to avoid unnecessary simulation time?

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    $\begingroup$ If you had the pdf of $X$, you could use importance sampling. Without the pdf, and only having access to a black-box simulation method, I can’t think of any good options. $\endgroup$ – guy Feb 14 at 14:49

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