Monte Carlo Simulation
Monte Carlo uses repeated random sampling to estimate mathematical quantities.
Basic Principle
Generate random inputs, compute outputs, aggregate over many trials. Law of large numbers ensures convergence.
Estimating Integrals
Sample-based integration works for high-dimensional problems. Error decreases as 1/√N.
Estimating π
Random points in unit square. Fraction inside quarter-circle approximates π/4.
Variance Reduction
Antithetic variates, control variates, stratified sampling, importance sampling improve accuracy.
Applications
Finance (option pricing), physics, engineering (reliability), ML (MCMC), optimisation (simulated annealing).
Summary
Monte Carlo is versatile for numerical estimation using random sampling.