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

1  Introduction to Monte Carlo

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Summary:

• Statistical computing is about solving statistical problems by combining human ingenuity with computing power.

• The Monte Carlo estimate of \(\Exg \phi(X)\) is \[ \widehat{\theta}_n^{\mathrm{MC}} = \frac{1}{n} \sum_{i=1}^n \phi(X_i) , \] where \(X_1, \dots, X_n\) are IID random samples from \(X\).

• Monte Carlo estimation typically gets more accurate as the number of samples \(n\) gets bigger.

Read more: Voss, An Introduction to Statistical Computing, Section 3.1 and Subsection 3.2.1.