CS(18) Monte Carlo Methods for Estimation

Monte Carlo methods are a powerful class of computational techniques used in statistical inference. They rely on repeated random sampling to compute numerical results, especially when traditional analytical solutions are difficult or impossible to obtain. The foundational idea is to use randomness to solve problems that might appear deterministic at first glance. Monte Carlo methods are particularly effective for:

1. Estimating probabilities and expectations.
2. Simulating complex stochastic systems.
3. Solving integrals or optimization problems numerically.

Foundation: Law of Large Numbers (LLN)

The Law of Large Numbers (LLN) forms the theoretical backbone of Monte Carlo methods. It states that the sample mean of a large number of independent and identically distributed (i.i.d.) random variables converges to the true population mean as the sample size increases. Mathematically, if X1, X2, …, Xn are i.i.d. random variables with mean 𝞵, then:

The LLN guarantees that using enough samples will yield an accurate estimate of the population mean. This principle…