Monte Carlo Simulation Case Study in MATLAB: Practical Application Examples

# 1. Fundamentals and Theory of Monte Carlo Simulation

Monte Carlo simulation is a numerical method that relies on random number generation to solve complex problems. Its core idea is to simulate random processes by repeatedly sampling randomly, and to infer the distribution or expected value of the target problem through statistical analysis of the sampling results.

The theoretical foundation of Monte Carlo simulation comes from probability theory and statistics. It uses random number generation functions to simulate random variables and describes the distribution of these random variables with probability distribution functions. Through repeated sampling, Monte Carlo simulation can generate a large number of random samples, and through statistical analysis of these samples, it obtains the approximate solution to the target problem.

# 2. Implementation of Monte Carlo Simulation in MATLAB

### 2.1 Syntax and Functions of Monte Carlo Simulation in MATLAB

#### 2.1.1 Random Number Generation Functions

MATLAB provides a variety of random number generation functions for generating random numbers with different distributions. The most commonly used functions include:

- `rand`: Generates uniformly distributed random numbers in the range [0, 1].
- `randn`: Generates standard normally distributed random numbers with a mean of 0 and a standard deviation of 1.
- `randsample`: Randomly samples a specified number of elements from a given vector or array.

**Code Block:**

% Generate 10 uniformly distributed random numbers
rand_nums = rand(1, 10);

% Generate 10 standard normally distributed random numbers
norm_nums = randn(1, 10);

% Randomly sample 5 elements from the range [1, 10]
sample_nums = randsample([1:10], 5);

**Logical Analysis:**

* The `rand` function generates a matrix of size 1×10, where each element is a uniformly distributed random number between 0 and 1.
* The `randn` function generates a matrix of size 1×10, where each element is a standard normally distributed random number.
* The `randsample` function randomly selects 5 elements from the range [1, 10] and returns a vector containing these 5 elements.

#### 2.1.2 Probability Distribution Functions

MATLAB also provides a variety of probability distribution functions for generating random numbers with specific distributions. The most commonly used functions include:

- `normrnd`: Generates normally distributed random numbers.
- `exprnd`: Generates exponentially distributed random numbers.
- `poisspdf`: Generates Poisson-distributed random numbers.

**Code Block:**

% Generate 10 normally distributed random numbers with a mean of 5 and a standard deviation of 2
norm_nums = normrnd(5, 2, 1, 10);

% Generate 10 exponentially distributed random numbers with a parameter of 3
exp_nums = exprnd(3, 1, 10);

% Generate 10 Poisson-distributed random numbers with a parameter of 5
pois_nums = poisspdf(5, [1:10]);

**Logical Analysis:**

* The `normrnd` function generates a matrix of size 1×10, where each element is a normally distributed random number with a mean of 5 and a standard deviation of 2.
* The `exprnd` function generates a matrix of size 1×10, where each element is an exponentially distributed random number with a parameter of 3.
* The `poisspdf` function generates a vector of size 1×10, where each element is the probability mass function value of a Poisson distribution with a parameter of 5.

### 2.2 Steps and Processes of Monte Carlo Simulation

#### 2.2.1 Problem Modeling and Parameter Setting

The first step in Monte Carlo simulation is to establish the mathematical model of the problem. This model should include all relevant variables, their distributions, and relationships. Once the model is established, values need to be specified for the parameters in the model. These parameters can be known constants or random variables.

**Code Block:**

% Define the problem model
model = @(x, y) x^2 + y^2;

% Define parameters
x_mean = 0;
x_std = 1;
y_mean = 0;
y_std = 1;

**Logical Analysis:**

* The `model` function defines the mathematical model of the problem, which calculates the sum of the squares of two random variables `x` and `y`.
* `x_mean`, `x_std`, `y_mean`, and `y_std` define the parameters of the normal distribution of `x` and `y`.

#### 2.2.2 Random Sampling and Calculation

Next, samples need to be randomly drawn from the parameter distributions in the model. These samples will be used to calculate the output of the model. The process of drawing samples is called Monte Carlo sampling.

**Code Block:**

% Randomly draw 1000 samples
num_samples = 1000;
x_samples = normrnd(x_mean, x_std, num_samples, 1);
y_samples = normrnd(y_mean, y_std, num_samples, 1);

% Calculat