\[ \newcommand{\Exg}{\operatorname{\mathbb{E}}} \newcommand{\Ex}{\mathbb{E}} \newcommand{\Ind}{\mathbb{I}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\ee}{\mathrm{e}} \]

3  Monte Carlo error I: theory

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Summary:

• The Monte Carlo estimator is unbiased.

• The Monte Carlo estimator has mean-square error \(\Var(\phi(X))/n\), so the root-mean-square error scales like \(1/\sqrt{n}\).

• The mean-square error can be estimated by \(S^2 / n\), where \(S^2\) is the sample variance of the \(\phi(X_i)\).

Read more: Voss, An Introduction to Statistical Computing, Subsection 3.2.2.