Monte Carlo Simulation

Simulation

Estimating Pi with Monte Carlo Simulation

Use random point generation to estimate the value of pi

Number of Points

Current estimate of pi: 0.000000

This simulation uses the Monte Carlo method to estimate pi. The left diagram shows the square and quarter circle, with green points inside the circle and red points outside. The right chart shows how the estimate converges to pi as more points are added.

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Understanding Monte Carlo Methods

Overview

Monte Carlo simulations are computational algorithms that use repeated random sampling to solve problems and obtain numerical results. These methods are particularly useful for optimizing, integrating and generating draws from complex probability distributions.

Key Concepts

1. Random Sampling

The foundation of Monte Carlo methods is generating random numbers and observing the fraction of numbers that obey some property or properties.

2. Law of Large Numbers

As the number of randomly generated points increases, the simulation result tends to converge to the true value, in accordance with the law of large numbers.

3. Convergence

Monte Carlo methods typically converge with $O(1/\sqrt{n})$ error, where n is the number of samples, making them relatively slow but reliable.

4. Error Estimation

The statistical error in Monte Carlo simulations can be estimated and reduced by increasing the number of iterations.

Mathematical Foundation

For our pi estimation example:

Area ratio method:

\frac{\text{Points in circle}}{\text{Total points}} \approx \frac{\pi r^2}{4r^2} = \frac{\pi}{4}

Therefore:

\pi \approx 4 \times \frac{\text{Points in circle}}{\text{Total points}}

The error in this estimate decreases as:

\text{Error} \propto \frac{1}{\sqrt{n}}

where n is the number of points sampled.

Applications

Physics: Particle diffusion, quantum chromodynamics, molecular dynamics
Finance: Option pricing, risk assessment, portfolio optimization
Engineering: Reliability analysis, circuit design, thermal analysis
Computer Graphics: Ray tracing, global illumination, image processing
Machine Learning: Bayesian inference, neural network initialization, reinforcement learning

Common Variations

Metropolis-Hastings Algorithm: For sampling from complex probability distributions
Gibbs Sampling: For sampling from multivariate distributions
Importance Sampling: For reducing variance in the estimates
Sequential Monte Carlo: For updating estimates as new data arrives

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