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

2  Uses of Monte Carlo

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Summary:

• The indicator \(\Ind_A(x)\) function of a set \(A\) is 1 if \(x \in A\) or 0 if \(x \notin A\).

• We can estimate a probability \(\mathbb P(X \in A)\) by using the Monte Carlo estimate for \(\Exg\Ind_A(X)\).

• We can estimate an integral \(\int h(x) \, \mathrm{d}x\) by using a Monte Carlo estimate with \(\phi(x)\,f(x) = h(x)\).

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