Question

4. Suppose g:[0,1]↦R is measurable and Lebesgue integrable. Let {U n ,n≥1} be iid uniform random variables and define X i =g(U i ). In what sense does ∑i=1nX i/n approximate ∫ 01g(x)dx ? (This offers a way to approximate the integral by Monte Carlo methods.) How would one guarantee a desired degree of precision?

Answer 1

The sum nX i approximates the integral ∫ 0 1 g(x)dx in the sense of convergence in probability. Increasing the number of random variables or adjusting the number of iterations n can guarantee a desired degree of precision.

Step-by-step explanation:

The sum ∑i=1nXi/n approximates the integral ∫01g(x)dx in the sense of convergence in probability. As n approaches infinity, the average of the random variables Xi approaches the Lebesgue integral of g(x) over the interval [0,1].

To guarantee a desired degree of precision, one can increase the number of random variables used in the sum or adjust the number of iterations n. The Law of Large Numbers ensures that as n increases, the average of the random variables will converge to the expected value of the Lebesgue integral.

Monte Carlo methods, such as this one, are commonly used in numerical integration and optimization problems.

Learn more about Monte Carlo methods