When a sum or integral of a variable cannot be computed directly, we can use Monte Carlo sampling to approximate the expected mean and variance of the random variable. For example, if s is a random variable, then $s = \sum_{x}p(x)f(x) = E_p[f(x)]$ or $s = \int p(x)f(x) = E_p[f(x)]$ We can approximate s by drawing n samples, and compute its estimator: $\hat{s} = \frac{1}{n}\sum_{i=1}^{n}f(x^{(i)})$ $\mathbb{E}[\hat{s}] = \frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[f(x^{(i)})] = \frac{1}{n}\sum_{i=1}^{n}s = s$ Similarly, we can obtain expected sampling variance. We can then compute the variance estimator:

$\mathrm{Var}[\hat{s_n}] = \frac{1}{n^2}\sum_{i=1}^{n}\mathrm{Var}[f(x^{(i)})] = \frac{\mathrm{Var}[f(x)]}{n}$

By Central Limit Theorem, the random variable thus follows Normal Distribution with mean s and variance $\frac{\mathrm{Var}[f(x)]}{n}$.