▶Start with Gaussians
Generate independent standard normal variables, which are easy to simulate.
The Monte Carlo Method: Core Idea
Creating complex distributions from simple random samples
Monte Carlo methods use repeated random sampling to produce numerical answers. A simple random variable can be transformed into one with a very different distribution.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: sample something easy, transform it, and let the empirical pattern reveal the target distribution.
One Concept · One Example
What to notice
- Repeated samples replace an unavailable formula or solver.
- A deterministic transformation can create a new distribution.
- More samples reduce empirical noise.
From a simple source to a new distribution
Transform and combine
The sum of their squares follows a chi-squared distribution with k degrees of freedom.
Watch a chi-squared distribution emerge
Choose the degrees of freedom and the sample count. The empirical histogram approaches the theoretical density as sampling increases.
Monte Carlo histogramχ²(1) density
Monte Carlo can approximate a complicated distribution by manipulating samples from simpler distributions.