Monte-Carlo Methods

Monte Carlo methods/experiments are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.

It is based on the Law of Large Numbers. This lecture notes seems to tell you that: https://people.math.umass.edu/~lr7q/ps_files/teaching/math456/lecture17.pdfA

This video that I didn’t watch by Shushen might be really good: https://www.youtube.com/watch?v=CmpWM2HMhxw&ab_channel=ShusenWang

$\mathbb{E}[f(x)]\approx \frac{1}{n}{\sum }_{i=1}^{n}f(x_i)X_i\sim piid$ We basically take the Mean.

The underlying concept is to use randomness to solve problems that might be deterministic in principle. Very useful when it is difficult or impossible to use other approaches.

Monte Carlo methods are mainly used in three problem classes:

1. Optimization
2. Numerical integration
3. Generating draws from a probability distribution