MATLAB Multivariate Normal Distribution Analysis: Exploring the Properties of Multivariate Normal Distribution

# Multivariate Normal Distribution Analysis in MATLAB: Exploring the Characteristics of Multivariate Normal Distribution

## 1. Theoretical Foundations of Multivariate Normal Distribution

### 1.1 Definition of Multivariate Normal Distribution

A multivariate normal distribution is a probability distribution that describes the joint distribution of multiple random variables. These random variables follow a normal distribution, and their covariance matrix is positive definite.

### 1.2 Probability Density Function

The probability density function for a multivariate normal distribution is given by:

f(x) = (2π)^(-p/2) |Σ|^(-1/2) exp(-1/2(x-μ)^T Σ^(-1) (x-μ))

Where:

- `x` is a p-dimensional random variable
- `μ` is a p-dimensional mean vector
- `Σ` is a p×p covariance matrix
- `|Σ|` is the determinant of `Σ`

## 2. Modeling Multivariate Normal Distribution in MATLAB

### 2.1 Probability Density Function of Normal Distribution

The normal distribution, also known as Gaussian distribution, is a continuous probability distribution with the following probability density function (PDF):

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

Where:

- `μ` is the mean of the distribution
- `σ` is the standard deviation of the distribution
- `π` is the mathematical constant Pi

In the case of a multivariate normal distribution, the PDF's dimension is the same as the number of variables. For a p-dimensional multivariate normal distribution, the PDF is:

f(x) = (1 / ((2π)^(p/2) * |Σ|^(1/2))) * exp(-(1/2) * (x - μ)' * Σ^(-1) * (x - μ))

Where:

- `x` is a p-dimensional vector representing the values of the variables
- `μ` is a p-dimensional vector representing the mean of the distribution
- `Σ` is a p×p covariance matrix
- `|Σ|` is the determinant of the covariance matrix

### 2.2 Covariance Matrix of Multivariate Normal Distribution

The covariance matrix describes the covariances between variables in a multivariate normal distribution. It is a symmetric matrix, where the diagonal elements represent the variances of the variables, and the off-diagonal elements represent the covariances between the variables.

The elements of the covariance matrix are:

Σ(i, j) = cov(X_i, X_j)

Where:

- `X_i` and `X_j` are two variables in the multivariate normal distribution
- `cov` is the covariance function

### 2.3 Generating Multivariate Normal Distribution in MATLAB

In MATLAB, the `mvnrnd` function can be used to generate random samples from a multivariate normal distribution. The syntax is:

X = mvnrnd(μ, Σ, n)

Where:

- `μ` is a p-dimensional vector representing the mean of the distribution
- `Σ` is a p×p covariance matrix
- `n` is the number of samples to generate

The `mvnrnd` function returns an `n x p` matrix, where each row represents a sample from the multivariate normal distribution.

**Code Block:**

% Define distribution parameters
mu = [0, 0];
Sigma = [1, 0.5; 0.5, 1];

% Generate 100 samples
X = mvnrnd(mu, Sigma, 100);

% Plot scatter diagram
scatter(X(:, 1), X(:, 2));
xlabel('X1');
ylabel('X2');
title('Scatter Plot of Multivariate Normal Distribution');

**Logical Analysis:**

This code generates 100 samples from a two-dimensional multivariate normal distribution. `mu` and `Sigma` represent the mean and covariance matrix of the distribution, respectively. The `mvnrnd` function returns a matrix `X` of size `100 x 2`, where each row corresponds to a sample. Finally, the `scatter` function plots the scatter diagram of the samples.

**Parameter Explanation:**

- `mu`: Mean vector of the distribution
- `Sigma`: Covariance matrix of the distribution
- `n`: Number of samples to generate

## 3. Statistical Inference of Multivariate Normal Distr