MATLAB Normal Distribution Random Variable Generation: Simulating Normal Distribution Random Variables

# Theoretical Foundations of the Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped probability density function. It is widely prevalent in nature and scientific research, describing the distribution of a large number of random variables.

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

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

Where μ represents the mean, and σ represents the standard deviation of the normal distribution. The mean indicates the central location of the distribution, while the standard deviation represents the dispersion of the distribution.

# Generating Normal Distribution Random Variables in MATLAB

### Definition and Properties of Normal Distribution Random Variables

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

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

Where μ is the mean, and σ is the standard deviation.

The normal distribution has the following properties:

***Symmetry:** The distribution curve is symmetric about the mean.
***Bell-shaped:** The distribution curve resembles a bell, with the highest point at the center and gradually decreasing on both sides.
***Asymptotic behavior:** The distribution curve falls rapidly near the mean and gradually approaches zero as it moves away from the mean.
***68-95-99.7 Rule:** Approximately 68% of the data falls within the range of μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ around the mean.

### MATLAB Functions for Generating Normal Distribution Random Variables

MATLAB provides two functions to generate normal distribution random variables:

#### 2.2.1 normrnd Function

**Syntax:**

X = normrnd(mu, sigma, m, n)

**Parameters:**

* mu: Mean of the normal distribution
* sigma: Standard deviation of the normal distribution
* m: Number of random variables to generate (rows)
* n: Number of random variables to generate (columns)

**Return Value:**

X: A matrix of size m×n containing normal distribution random variables.

**Code Block:**

% Generate normal distribution random variables with mean 0 and standard deviation 1
X = normrnd(0, 1, 1000, 1);

% View the generated random variables
histogram(X);
xlabel('Value');
ylabel('Frequency');
title('Normal Distribution Random Variables');

**Logical Analysis:**

* The first line of code uses the normrnd function to generate 1000 normal distribution random variables with a mean of 0 and a standard deviation of 1, and stores them in the variable X.
* The second line of code uses the histogram function to plot the histogram of the random variables, with the x-axis representing values and the y-axis representing frequencies.
* The third line of code sets the x-axis label to "V