Monte Carlo Simulation

A Monte Carlo method generates samples from one or more distributions and computes an outcome for each draw. Averages, quantiles or event frequencies approximate properties of the modeled system. The method can handle relationships too complicated for an analytic calculation, provided the sampling procedure represents them correctly. Repeated draws introduce sampling error that can be estimated and reduced. Variance-reduction techniques and quasi-Monte Carlo sequences alter how samples cover the space, but have their own assumptions and interpretation. Simulation is therefore both a model of uncertainty and a numerical estimation procedure, with separate questions about model validity and estimator precision.

Define the target quantity and the uncertain inputs, including correlations and constraints. Check how each distribution was justified and avoid sampling related variables independently without reason. Control random seeds for reproducibility, assess convergence across sample budgets and quantify Monte Carlo error where possible. Use sensitivity analysis to identify influential assumptions and compare against a simple case with a known result. The output should describe a distribution of outcomes and the assumptions producing it, rather than one apparently exact number from an arbitrary number of draws.

More draws improve numerical precision but cannot repair an unrealistic distribution or omitted dependency. Rare events may require specialized sampling; ordinary simulation can miss them and give false reassurance. A reproducible random seed is useful for debugging, not evidence of correctness. Quasi-Monte Carlo sampling differs from independent random sampling, so ordinary error formulas may not apply unchanged. Check impossible simulated states, tail behavior and sensitivity, and distinguish uncertainty about inputs from uncertainty introduced by finite simulation effort.