Variance for Hit-and-Miss Monte Carlo is given by $Var(\theta)=\frac{\Theta*(1-\Theta)}{N}$ where $\theta$ is the estimated probability of Hit and N is the number of simulations. Can someone explain why? And what will be the variance when Importance Sampling is used?
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$\begingroup$ How do you define importance sampling in connection with hit-and-miss Monte Carlo? With no restriction, the minimal variance of an importance sampling estimate is zero. $\endgroup$– Xi'anCommented Apr 12, 2013 at 20:14
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The first is just the variance of $\hat{p} = X/n$ in a binomial.
If $X$ is the number of "hits" in $n$ trials,
$$\newcommand{\Var}{\operatorname{Var}}\Var(X) = n\theta(1-\theta)$$
(Wikipedia on the Binomial Distribution).
So
$$\Var(X/n) = \frac{1}{n^2} n\theta(1-\theta) = \frac{\theta(1-\theta)}{n}.$$
For a comparison of the variance between the plain binomial case and importance sampling, see this section of the Wikipedia article on importance sampling.
