Question

Compute \(\pi\) by a random sum. Let \(x_{0}, \ldots, x_{N}\) be \(N+1\) uniformly distributed random numbers between 0 and \(1 .\) Explain why the random sum \(S_{N}=\sum_{i=0}^{N} 2\left(1-x_{i}^{2}\right)^{-1}\) is an approximation to \(\pi\). (Hint: Interpret the sum as Monte Carlo integration and compute the corresponding integral exactly by hand.) Make a program for plotting \(S_{N}\) versus \(N\) for \(N=10^{k}, k=\) \(0,1 / 2,1,3 / 2,2,5 / 2, \ldots, 6\). Write out the difference between \(S_{10^{6}}\) and pi from the math module. Name of program file: MC_pi_plot.py.

Short answer

The sum approximates \(\pi\) using Monte Carlo integration of an appropriate integral.

Step-by-step solution

Understand the Monte Carlo Method

Monte Carlo integration is a statistical method used to approximate the value of an integral by using the average value of random samples. Here, we're using this method to approximate \(\pi\). The idea is to simulate the integral by generating random numbers between 0 and 1, which act as sample points.

Set Up the Integral

Our goal is to approximate \(\pi\) using the equation \[ \pi = \int_{0}^{1} \frac{2}{1-x^2} \, dx. \] This integral represents multiplying the probability density function with 2 and integrating across the interval [0, 1].

Compute the Integral Analytically

To compute the integral, we have \[ \int_{0}^{1} \frac{2}{1-x^2} \, dx \]. Substituting \( u = x \) and \( du = dx \), we realize this is related to \( \arctanh(x) \). Solving gives: \[ \int_{0}^{1} \frac{2}{1-x^2} \, dx = 2 \times \frac{1}{2} \left( \ln\left(\frac{1+x}{1-x}\right) \bigg|_{0}^{1} \right) = \pi. \] Thus, the integral itself evaluates to \(\pi\).

Create the Random Sum Expression

The sum \( S_{N} = \sum_{i=0}^{N} 2 \left( 1 - x_i^2 \right)^{-1} \) effectively becomes the numerical estimation of the integral for a large \( N \). Each call to the sum represents a sampled value from the distribution over the interval [0, 1].

Python Program for Plotting

Develop a Python program named `MC_pi_plot.py` using libraries such as `numpy` and `matplotlib`. Use the following structure:1. Import necessary modules: `import numpy as np` and `import matplotlib.pyplot as plt`.2. Define a function `compute_S_N(N)` that uses the sum formula \( S_N \).3. Populate values of \( N \) for specified \( k \).4. Plot \( S_N \) against \( N \) using `plt.plot()` and display using `plt.show()`.

Compute and Compare for Large N

Calculate \( S_{10^6} \) using previously written code by calling `compute_S_N(10**6)`. Compare the result to \(\pi\) using Python's `math.pi` to find the difference: \text{difference} = |S_{10^6} - \pi|. This shows how close the approximation is to the actual value.

Key concepts

Random Number Generation

Random number generation is a process used in simulations and calculations to produce numbers that lack any predictable pattern.
In the context of Monte Carlo Integration, these numbers are crucial because they simulate the behavior of a variable in a model by generating samples over a specific interval.

• In generating \(x_{0}, \ldots, x_{N}\), we rely on a function that can produce a uniform distribution of numbers between 0 and 1.
• This approach replicates the randomness necessary for a statistical simulation environment, offering each number equal probability of being chosen within the specified range.
• The quality of these random numbers can significantly impact the accuracy of results, which is why robust algorithms are pivotal, especially in scientific programming languages or libraries like Python's `numpy`.

This randomness underpins the Monte Carlo method, allowing for realistic predictions and approximations in complex calculations.

Integration Techniques

Integration is a fundamental concept in calculus, often used to find areas under curves.
Monte Carlo Integration is a numerical technique that leverages random sampling to estimate the value of a definite integral.

• Unlike traditional analytical integration, which uses algebraic anti-derivatives, Monte Carlo integration uses averages of random samples to approximate an integral.
• This technique is particularly useful when dealing with high-dimensional spaces or functions that do not have simple antiderivatives.

In our exercise, we use this method to approximate \( \pi \) by integrating over a function. Think of simulating random darts hitting within a unit circle and using the ratio of darts inside the circle to total darts to approximate \( \pi \). This technique effectively turns a complex task into a manageable sampling activity.

Numerical Methods

Numerical methods are algorithms for approximating solutions to mathematical problems.
They are essential when an exact solution is impossible or impractical to determine analytically.

• Monte Carlo Integration is one such numerical method, offering a probabilistic approach to estimating integrals.
• Unlike deterministic methods, Monte Carlo doesn't guarantee the same output upon each execution, due to its reliance on randomness.
• This variability can be minimized by using a large number of samples, as more samples typically lend a more accurate approximation.

The key advantage lies in Monte Carlo's flexibility and relative ease of application, especially where traditional means fail. It's a powerful tool, providing workable solutions for both simple and complex mathematical models.

Python Programming

Python is a versatile language, particularly strong in its simplicity and wide array of libraries that support scientific computation.
For simulation tasks like Monte Carlo integration, Python provides everything necessary to orchestrate and visualize complex calculations:

• The numpy library is essential for efficient array manipulations and computational tasks. It allows creation and operations on arrays of random numbers, critical for generating samples in our Monte Carlo simulation.
• Matplotlib is another indispensable tool, offering plotting capabilities to visualize the relationship between sample size and calculated results.

In our particular exercise, writing a Python script isn’t just about computing values; it's about creating a procedural framework that can be adapted and adjusted for further experimentation, making Python an excellent choice for educational programs and real-world applications.

Pi Calculation

The calculation of \(\pi\) is not only a classic problem in mathematics but also a practical demonstration of numerical methods' power.
With Monte Carlo integration, \(\pi\) is approximated as a sum over randomly generated variables – each contributing to a statistical interpretation of the circle's area.

• The function \( S_{N} = \sum_{i=0}^{N} 2(1-x_i^2)^{-1} \) provides an iterative approximation for \( \pi \), using the cumulative influence of random inputs to stabilize to the true value with increasing \( N \).
• This process encapsulates the elegance of Monte Carlo methods, effectively replacing complex integration with simple summation and averaging.
• Although the method converges more slowly than some direct algorithms, its simplicity and flexibility make it a powerful method for acquiring \(\pi\) to a high degree of accuracy in a statistical manner.

Understanding this connection between randomness, summation, and mathematical constants like \(\pi\), really highlights the far-reaching applications of numerical methods in scientific computations.