 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/continuous-mathematics/5/a/iv/solution

A convenient choice for this irregular region is Monte Carlo integration. The region is contained in

[- 1, 1]^{2}

. For independent uniform points

U_{j}

in that square, let

I_{N} = \frac{4}{N} \sum_{j = 1}^{N} 1_{{U_{j} \in R}} exp (e^{(U_{j})_{1}} + e^{(U_{j})_{2}} + e^{(U_{j})_{1} (U_{j})_{2}}) .

The indicator random variable incorporates the curved boundary directly. This estimator is unbiased; the integrand is bounded on the square, so its variance is finite. The variance of a sum of independent terms gives root mean square error $O (N^{- 1/2})$ in big O notation. This is a stochastic error rate. Applying a rectangular Simpson's rule to the indicator-weighted integrand would lose the smoothness required for its fourth-order error bound.

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