Random Number Generation, Sampling, and Simulation with Julia:

Week 5: Monte Carlo Simulation in Julia

Lesson Plan: Monte Carlo Simulation in Julia

Class 1: Introduction to Monte Carlo Methods

Objective:

Understand the fundamentals of Monte Carlo methods, including their history and key concepts.

Materials:

  • Computer with Julia installed
  • Internet access for research
  • Whiteboard or presentation slides

Agenda:

  1. Introduction (10 minutes)
    • Define Monte Carlo methods.
    • Discuss the historical development and significance.
  2. Key Concepts (20 minutes)
    • Explain random number generation and probability distributions.
    • Introduce the Law of Large Numbers.
  3. Real-world Examples (15 minutes)
    • Estimating the value of π using random sampling.
    • Simulating stock prices.

Activities:

  • Group discussion on the importance of randomness in simulations.
  • Q&A session to clarify concepts.

Class 2: Implementing Basic Monte Carlo Simulations

Objective:

Learn to implement basic Monte Carlo simulations using the Julia programming language.

Materials:

  • Computer with Julia installed
  • Jupyter Notebook or any Julia IDE

Agenda:

  1. Introduction to Julia (10 minutes)
    • Overview of Julia’s capabilities for numerical and statistical computing.
  2. Code Example: Estimating the Value of π (35 minutes)
using Random

function monte_carlo_pi(n::Int)
    inside_circle = 0
    for i in 1:n
        x, y = rand(), rand()
        if x^2 + y^2 <= 1.0
            inside_circle += 1
        end
    end
    return (inside_circle / n) * 4
end

println(monte_carlo_pi(10000))

Activities:

  • Implement the provided code example.
  • Experiment with different sample sizes and observe the effect on accuracy.
  • Compare results with classmates.

Class 3: Applications of Monte Carlo Simulations in Various Fields

Objective:

Explore various applications of Monte Carlo simulations in different fields.

Materials:

  • Computer with internet access for research
  • Presentation slides or whiteboard

Agenda:

  1. Applications Overview (20 minutes)
    • Discuss applications in finance, physics, engineering, and computer science.
  2. Examples and Case Studies (25 minutes)
    • Finance: Risk assessment and portfolio management.
    • Physics: Particle transport and diffusion.
    • Engineering: Reliability analysis and optimization.
    • Computer Science: Rendering and game development.

Activities:

  • Group discussions on how Monte Carlo methods can be applied in students’ fields of interest.
  • Research and present a case study where Monte Carlo simulations have been used effectively.

Class 4: Evaluating Simulation Accuracy and Convergence

Objective:

Learn techniques for evaluating the accuracy and convergence of Monte Carlo simulations.

Materials:

  • Computer with Julia installed
  • Jupyter Notebook or any Julia IDE
  • Plotting library installed (e.g., Plots.jl)

Agenda:

  1. Accuracy and Convergence Techniques (20 minutes)
    • Discuss statistical measures such as variance, standard error, and confidence intervals.
  2. Code Example: Evaluating Convergence (25 minutes)
function simulate_convergence(trials::Int)
    estimates = Float64[]
    for n in 1:trials
        push!(estimates, monte_carlo_pi(n * 1000))
    end
    return estimates
end

using Plots
plot(simulate_convergence(100), title="Convergence of Monte Carlo π Estimation", xlabel="Trials", ylabel="Estimate of π")

Activities:

  • Implement the convergence evaluation code and plot the results.
  • Analyze the impact of sample size on simulation accuracy and convergence.
  • Discuss findings and share insights with classmates.

Assessment:

  • Participation in discussions and activities.
  • Successful implementation and experimentation with provided code examples.
  • Group presentations on Monte Carlo applications.
  • Evaluation of simulation accuracy and convergence.

Additional Resources:

  • Books, articles, and online tutorials on Monte Carlo methods.
  • Julia documentation and community resources for further learning.

Using this lesson plan, students will have ample time to delve deep into the world of Monte Carlo simulations and gain hands-on experience with the Julia programming language.

Introduction to Monte Carlo Methods

A short tutorial on Monte Carlo simulation using Julia could cover the following key concepts and examples, aiming for a balance between breadth and depth suitable for a beginner:

I. Introduction to Monte Carlo Methods (15 mins)

  • What are Monte Carlo methods? Briefly explain the core idea of using randomness for problem-solving. Mention applications in finance, physics, etc.
  • Why use Julia for Monte Carlo? Highlight Julia’s speed, ease of use, and strong support for scientific computing.
  • Basic building blocks: Random number generation in Julia (rand(), randn(), distributions from Distributions.jl).

II. Estimating Pi with Monte Carlo (30 mins)

  • A classic example: Explain how to estimate pi by randomly throwing darts at a square containing a circle.
  • Julia implementation: Write the code step by step, explaining how to generate random points, check if they fall within the circle, and calculate the estimate of pi. Emphasize vectorization for efficiency.
  • Visualizing the results: Use Plots.jl to visualize the dart throws and the convergence of the pi estimate.

III. Monte Carlo Integration (45 mins)

  • The concept: Explain how Monte Carlo can be used to approximate definite integrals.
  • Example 1: Integrating a simple function (e.g., x^2) over a given interval. Implement in Julia.
  • Example 2: A slightly more complex example, perhaps involving a non-standard function or a multi-dimensional integral. Show how to handle this in Julia.
  • Variance reduction (optional): Briefly introduce the concept of variance reduction techniques (e.g., importance sampling, control variates) to improve the accuracy of Monte Carlo estimates. A very basic example is sufficient.

IV. Simulating a Simple Stochastic Process (45 mins)

  • Introduction to stochastic processes: Give a very brief overview (e.g., a random walk).
  • Example: Simulating a simple random walk in 1D or 2D.
  • Julia implementation: Show how to simulate the random walk step by step, using loops or vectorization.
  • Visualization: Plot the path of the random walk.

V. Briefly Touch on More Advanced Topics (15 mins)

  • Mention other applications: Briefly discuss Monte Carlo methods in finance (option pricing), Bayesian statistics, and other fields.
  • Packages: Briefly introduce relevant Julia packages (e.g., Distributions.jl for more distributions).
  • Further learning: Point to resources for further study (books, online courses, documentation).

Example Code Snippets (Illustrative):

# Estimating Pi
using Plots
n_darts = 10000
x = rand(n_darts)
y = rand(n_darts)
inside_circle = x.^2 + y.^2 .<= 1
pi_estimate = 4 * sum(inside_circle) / n_darts
scatter(x, y, color = ifelse.(inside_circle, :red, :blue), markersize = 2, aspect_ratio = 1,
        xlabel = "x", ylabel = "y", title = "Estimating Pi")
println("Pi estimate: ", pi_estimate)

# Monte Carlo Integration
f(x) = x^2
a = 0
b = 1
n_samples = 10000
x = rand(n_samples) * (b - a) + a  # Generate random numbers in [a, b]
integral_estimate = (b - a) * mean(f.(x))
println("Integral estimate: ", integral_estimate)

Tutorial Delivery:

  • Could be delivered as a Jupyter notebook, allowing for a mix of code, explanations, and visualizations.
  • Interactive elements could be added (e.g., sliders to change the number of samples).
  • A live coding session would be very beneficial.

Key Considerations:

  • Keep it beginner-friendly: Avoid overly complex mathematical details.
  • Focus on clear explanations and practical examples.
  • Emphasize Julia’s strengths: Speed and ease of use.
  • Encourage experimentation: Have exercises for the participants.

This structure provides a solid foundation for a short, effective tutorial on Monte Carlo simulation using Julia. Remember to adjust the time allocation and the complexity of the examples based on the audience’s background.