The Power Tool for Quantifying Uncertainty: Monte Carlo Simulation in MATLAB

# The Power of Uncertainty Quantification: Monte Carlo Simulation in MATLAB

## 1. Introduction to Monte Carlo Simulation

Monte Carlo simulation is a powerful technique widely used to solve complex problems and perform probabilistic modeling. It relies on random number generation and probability theory to approximate the solution to mathematical problems by simulating a large number of random events.

The advantages of Monte Carlo simulation include:

- **Applicability to Complex Problems:** It can solve complex problems that traditional methods struggle with, such as high-dimensional integration and nonlinear equations.
- **Probabilistic Modeling:** It can simulate random events and uncertainties, providing decision-making insights based on probability.
- **Parallel Computing:** Monte Carlo simulation lends itself easily to parallelization, thus enhancing performance in modern multicore computing environments.

## 2. Theoretical Foundations of Monte Carlo Simulation in MATLAB

### 2.1 Principles and Advantages of the Monte Carlo Method

The Monte Carlo method is a numerical technique based on probability and statistics used to solve complex problems. It works by generating a vast number of random samples and approximating the solution based on the statistical characteristics of these samples.

The benefits of the Monte Carlo method are:

- **Universality:** It can be applied to solve a wide variety of mathematical and physical problems.
- **Robustness:** It is insensitive to nonlinearities and discontinuities in the problem.
- **Parallelism:** Sample generation can be parallelized, increasing computational efficiency.

### 2.2 Random Number Generation and Distribution Functions

In Monte Carlo simulation, random number generation is crucial. MATLAB offers several random number generators, including:

rand() % Generates uniformly distributed random numbers between 0 and 1
randn() % Generates normally distributed random numbers with a mean of 0 and a standard deviation of 1
poissrnd(lambda) % Generates Poisson distributed random numbers with an average rate of occurrence lambda

Additionally, MATLAB provides distribution functions to map random numbers to specific distributions. For example:

normcdf(x, mu, sigma) % Calculates the cumulative distribution function of a normal distribution with mean mu and standard deviation sigma
unifcdf(x, a, b) % Calculates the cumulative distribution function of a uniform distribution with a minimum value a and a maximum value b

**Code Block Logic Analysis:**

* The `rand()` function generates a uniformly distributed random number between 0 and 1.
* The `randn()` function generates a normally distributed random number with a mean of 0 and a standard deviation of 1.
* The `poissrnd()` function generates a Poisson distributed random number with an average occurrence rate of lambda.
* The `normcdf()` function calculates the cumulative distribution function of a normal distribution, where mu is the mean and sigma is the standard deviation.
* The `unifcdf()` function calculates the cumulative distribution function of a uniform distribution, where a is the minimum value and b is the maximum value.

**Paramete