MATLAB Normal Distribution Monte Carlo Simulation: Exploring Random Simulation of Normal Distribution

# Introduction to Normal Distribution and Monte Carlo Simulation in MATLAB

## 1. Overview of Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by a bell-shaped curve. It is ubiquitous in nature and statistics, describing the distribution of numerous random variables.

The probability density function of the normal distribution is given by the following formula:

f(x) = (1 / (σ√(2π))) * e^(-(x-μ)² / (2σ²))

Where:

* μ is the mean of the distribution, indicating the central tendency of the data.
* σ is the standard deviation of the distribution, representing the dispersion of the data.

## 2. Principles of Monte Carlo Simulation

Monte Carlo simulation is a numerical method based on random number generation to solve complex problems. It approximates the expected value or distribution of a target function through repetitive random sampling and calculation.

### 2.1 Random Number Generation

The core of Monte Carlo simulation is random n***mon methods of random number generation include:

- **Linear congruential method:** Using a linear congruential equation to generate random numbers, e.g., `x[n+1] = (a * x[n] + c) % m`, where `a`, `c`, `m` are constants.
- **Mersenne Twister algorithm:** An efficient random number generation algorithm that uses linear feedback shift registers to produce random numbers.
- **Pseudorandom number generators (PRNGs):** Deterministic algorithms that generate sequences that appear random but are actually predictable.

### 2.2 Probability Distribution Sampling

In Monte Carlo simulation, ***monly used probability distributions include:

- **Uniform distribution:** Random numbers are uniformly distributed within a given range.
- **Normal distribution:** Random numbers follow a normal distribution, i.e., a bell-shaped curve.
- **Poisson distribution:** Random numbers follow a Poisson distribution, used to represent the number of events occurring within a fixed interval of time or space.

Sampling methods vary according to the probability distribution, such as:

- **Uniform distribution:** Random numbers are generated directly within the given range.
- **Normal distribution:** Random numbers are generated using normal distribution functions like `randn` or `normrnd`.
- **Poisson distribution:** Random numbers are generated using Poisson distribution functions like `poissrnd`.

### 2.3 Monte Carlo Integration

Monte Carlo integration is a method of using Monte Carlo simulation to solve integrals. It estimates the integral value by randomly sampling the integrand within a given region.

**Algorithm:**

1. Generate `N` random points `(x[i], y[i])` within the given region.
2. Calculate the function value `f(x[i], y[i])` for each point.
3. Estimate the integral value: `I ≈ (1/N) * ∑[f(x[i], y[i])]`

**Parameter Explanation:**

- `N`: The number of random sampling points.
- `f(x, y)`: The function to be integrated.

**Code Example:**

% Define the function to be integrated
f = @(x, y) x.^2 + y.^2;

% Define the boundaries of the region
xmin = 0;
xmax = 1;
ymin = 0;
ymax = 1;

% Random sampling
N = 10000;
x = xmin + (xmax - xmin) * rand(N, 1);
y = ymin + (ymax - ymin) * rand(N, 1);

% Calculate function values
z = f(x, y);

% Estimate the integral value
I = (1/N) * sum(z);

disp(['Estimated integral value: ', num2str(I)]);

**Logical Analysis:**

This code implements the Monte Carlo integration algorithm. It first defines the function `f(x, y)` to be integrated, then specifies the boundaries of the integration region. Next, it generates `N` random points and calculates the function values for each point. Finally, it estimates the integral value based on the results of the random sampling.

# 3.1 Normal Distribution Functions in MATLAB

MATLAB offers a variety of normal distribution functions for generating normal distribution random numbers, calculating probability density, and cumulative distribution functions.

**Generating Normal Distribution Random Numbers**

The `normrnd` function is used to generate normal distribution random numbers:

% Normal distribution with mean 0, standard deviation 1
x = normrnd(0, 1, 1000, 1);

**Calculating Probability Density**

The `normpdf` function calculates the probability density of a normal distribution:

% Calculate the probability density at x=0
p = normpdf(0, 0, 1);

**Calculating Cumulative Distribution Function**

The `normcdf` function calculates the cumulative distribution function of a normal distribution:

% Calculate the cumulative distribution function at x=0
p = normcdf(0, 0, 1);

### 3.2 Generating Normal Distribution Random Numbers

In MATLAB, the `normrnd` function