Which of the following best describes Monte Carlo simulation?

Question

BlackTom AI · Accepted Answer

Monte Carlo simulation approximates expectations (integrals) by averaging simulated outcomes from a specified probabilistic model.

1) Relevant knowledge points and concepts
- Law of large numbers (LLN): the sample average of iid draws converges to the true expectation as sample size n → ∞.
- Central limit theorem (CLT): the sampling distribution of the average is approximately normal for large n, enabling error/CI estimates.
- Monte Carlo estimator: generate samples from a distribution, evaluate the quantity of interest for each sample, and average.

2) Step-by-step reasoning process
- Suppose we want E[g(X)] for a random variable X with known or simulatable distribution.
- Draw independent samples X1, X2, …, Xn from the distribution of X.
- Compute Yi = g(Xi) for each draw.
- Use the Monte Carlo estimator μ̂ = (1/n) Σ_{i=1}^n Yi to approximate E[g(X)].
- By LLN, μ̂ → E[g(X)] as n increases; by CLT, μ̂ ≈ Normal(E[g(X)], Var(Y)/n) for large n.

3) Calculations and intermediate steps (example)
- Example: X ~ Uniform(0,1), want E[X^2] (true value = 1/3).
- Generate n samples xi, compute yi = xi^2.
- Estimate μ̂ = (1/n) Σ yi. The estimator is unbiased: E[μ̂] = E[X^2] = 1/3.
- Variance of the estimator: Var(μ̂) = Var(X^2)/n = (E[X^4] − (E[X^2])^2)/n = (1/5 − (1/3)^2)/n = (1/5 − 1/9)/n = (4/45)/n.
- As n increases, Var(μ̂) decreases, improving accuracy.

4) Why the correct answer is correct
- Monte Carlo methods are precisely about approximating expectations or integrals by averaging simulated outcomes from the relevant probability model. This matches option b.

5) Why the other options are incorrect
- a: “Estimating parameters by minimizing variance” is not a definition of Monte Carlo (though variance reduction techniques exist within Monte Carlo).
- c: “Sorting the data” describes ordering or computing order statistics, not Monte Carlo simulation.
- d: “Estimating parameters by maximizing likelihood” defines maximum likelihood estimation (MLE), a different method (though MLE can be used alongside Monte Carlo in some contexts).

6) Conclusion
- Therefore the best description is b. This matches the provided correct answer: b. Approximating an expectation by averaging simulated outcomes.

Which of the following best describes Monte Carlo simulation?单项选择题

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