Monte Carlo Simulation

Probability · Axiom Academy

Estimating probabilities and expectations through random sampling with law of large numbers justification 1. Basic Monte Carlo Estimation To estimate E[g(X)] for some function g and random variable X, we draw independent samples X₁, X₂, ..., Xₙ from the distribution of X and compute the sample mean: This estimator is unbiased: E[g̃ₙ] = E[g(X)]. The Law of Large Numbers guarantees that g̃ₙ → E[g(X)] as n → ∞. 2. Estimating Probabilities via Indicators To estimate P(X ∈ A), use indicator functions. Let I(Xᵢ ∈ A) = 1 if Xᵢ ∈ A and 0 otherwise. The sample proportion gives an estimate: Since E[I(X ∈ A)] = P(X ∈ A), this is an unbiased estimator. The Central Limit Theorem provides error bounds: the standard error is approximately √[p(1-p)/n]. 3. Variance Reduction and Convergence The accuracy of Monte Carlo estimates improves at rate O(1/√n). By the CLT, the standardized error converges to a normal distribution: Variance reduction techniques (antithetic variables, importance sampling, control variates) can reduce σ² and improve efficiency without increasing n.

This is the written version of the interactive lesson above. See the full Probability course.