Fundamentals of Monte Carlo
What is Monte Carlo?
- Named at Los Alamos in 1940’s after the casino.
- Any method which uses (pseudo)random numbers> as an essential part of the algorithm.
- A method for doing highly dimensional integrals by sampling the integrand.
- Often a Markov chain, called Metropolis MC.
Stochastic - not deterministic!
- Simple example: Buffon’s needle - Monte Carlo determination of π
- MC is advantageous for high dimensional integrals -the best general method
- Improved Numerical Integration
- Other reasons to do Monte Carlo
- Probability Distributions
- Mappings of random variables
- What is Mapping Doing?
- Interpreting the Mapping
- Example: Drawing from Normal Gaussian
- Reminder: Gauss’ Central Limit Theorem
- Cumulants: κn Mean = κ1 Variance= κ2 Skewness = κ3 Kurtosis= κ4
- Approach to normality
- Conditions on Central Limit Theorem
- 2d histogram of occurrences of means