Past exam of the simulation and statistical programming course of the University of Oxford / 2023 / 1 / a / i / Solution

Generate independent and identically distributed random variables $X_1,\dots ,X_n$ with probability density function $p$, evaluate $\varphi (X_i)$ for each draw, and return the Monte Carlo estimatorLinearity of expectation makes this an unbiased estimator: $\mathbb{E}{\hat{\theta }}_n^{\mathrm{MC}}=\theta$. Independence and the specified variance give $\mathrm{Var}({\hat{\theta }}_n^{\mathrm{MC}})=1/n$. Its Monte Carlo standard error is therefore $n^{-1/2}$, and the strong law of large numbers gives convergence to $\theta$ almost surely.