Monte Carlo Method
The Monte Carlo method approximates deterministic problems—such as integration, optimization, and probability estimation—by random sampling, converging at O(1/√N).
Purpose
To approximately compute high-dimensional integrals and complex probability distributions.
Mechanism
Approximate the expectation with the statistical average of N random samples: ∫f(x)p(x)dx ≈ (1/N)∑f(x_i). The law of large numbers guarantees convergence; the central limit theorem gives error O(1/√N), independent of dimension.
Use cases
- High-dimensional integration
- Probability simulation
- Physics computation