Asked to compute estimator for the following function,
$\theta = \int_0^\infty e^{-x^2}$
which can be solved by transforming the limits to 0 to 1 and solving the following expectation using Monte Carlo,
$$ E[1/U \space e^{-1(-1+1/U)^2}]$$
where $U$ is uniformly distributed
The estimator is $$\hat\theta = \frac 1N \sum_0^n \frac 1U \space e^{-1(-1+1/U)^2}. $$
How do I go about computing the sample variance and then the error? Are there any formulas to choose the best $N$?
