Monte Carlo Integration

Probability · Axiom Academy

EXAMPLE Monte Carlo Integration Estimating π using random sampling and understanding convergence Excellent work! You've mastered Monte Carlo integration. Here's what we learned: Monte Carlo Principle: Convert integrals into expectations by random sampling. The sample average converges to the true expected value by the Law of Large Numbers. π Estimation: By sampling uniform points in a square and counting those inside the inscribed circle, we estimate π/4 as the ratio of areas, then multiply by 4. Convergence Rate: The error decreases as 1/√n by the Central Limit Theorem. To halve the error, we need 4 times as many samples. Variance Reduction: Techniques like importance sampling, stratified sampling, and control variates can dramatically improve convergence by reducing the variance of the estimator. Monte Carlo methods are powerful for high-dimensional integration where traditional numerical methods fail. They trade exactness for simplicity and scalability!

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