# Any trick to evaluate $c\times \textrm{Pr}\{X < a\}$?

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?

• 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. – guy Feb 14 at 14:49