Summary of Monte Carlo Simulations for Power Point

Slide 1: Title

The Enigmatic Elegance of Monte Carlo Simulations: Unveiling the Secrets of Randomness in Complex Systems

Slide 2: Introduction

• Monte Carlo simulation: powerful numerical technique
• Applicable in finance, physics, engineering, and more

Slide 3: Historical Background

• Pioneered by Stanislaw Ulam, John von Neumann, and Nicholas Metropolis
• Developed while studying nuclear chain reactions

Slide 4: Theoretical Foundations

• Law of Large Numbers: sample mean converges to population mean
• Random Number Generation: key to simulating stochastic systems

Slide 5: Mechanics of the Method

1. Formulate a probabilistic model
2. Perform sampling and simulation
3. Analyze and interpret results

Slide 6: Probabilistic Model

• Accurately reflects stochastic behavior of system
• Essential for successful simulation

Slide 7: Sampling and Simulation

• Generate random samples based on probability distributions
• Simulate various possible outcomes

Slide 8: Analysis and Interpretation

• Examine results for statistical data
• Determine mean, variance, and percentiles

Slide 9: Applications

• Finance: option pricing, portfolio optimization, risk assessment
• Physics: radiation transport, statistical mechanics

Slide 10: Conclusion

• Revolutionized understanding of randomness in complex systems
• Unlocks potential for new discoveries and insights

Python Code of Monte Carlo Simulations

Here’s a simple example of a simulation in Python, estimating the value of pi:

import random
import math

def monte_carlo_simulation_pi(num_points):
    points_inside_circle = 0

    for _ in range(num_points):
        x = random.uniform(-1, 1)
        y = random.uniform(-1, 1)
        distance_from_origin = math.sqrt(x**2 + y**2)

        if distance_from_origin <= 1:
            points_inside_circle += 1

    pi_estimate = 4 * points_inside_circle / num_points
    return pi_estimate

num_points = 1000000
pi_estimate = monte_carlo_simulation_pi(num_points)
print(f"Estimated value of pi using {num_points} points: {pi_estimate}")

import random
import math

def monte_carlo_simulation_pi(num_points):
    points_inside_circle = 0

    for _ in range(num_points):
        x = random.uniform(-1, 1)
        y = random.uniform(-1, 1)
        distance_from_origin = math.sqrt(x**2   y**2)

        if distance_from_origin <= 1:
            points_inside_circle  = 1

    pi_estimate = 4 * points_inside_circle / num_points
    return pi_estimate

num_points = 1000000
pi_estimate = monte_carlo_simulation_pi(num_points)
print(f"Estimated value of pi using {num_points} points: {pi_estimate}")

This Python code demonstrates a Monte Carlo simulation to estimate the value of pi. The simulation generates random points within a square with side length 2, centered at the origin, and counts the number of points that fall inside a circle with radius 1, also centered at the origin. The estimated value of pi is calculated based on the ratio of the points inside the circle to the total number of points.

Reference

WIKIPEDIA: Monte Carlo Simulation

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