MATLAB Normal Distribution Correlation Analysis: Exploring the Association between Normally Distributed Variables

# Normal Distribution Correlation Analysis in MATLAB: Exploring the Association Between Variables

## 1. Theoretical Basis of Normal Distribution in MATLAB

The normal distribution, also known as the Gaussian distribution, is a common probability distribution characterized by its probability density function:

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

Here, μ represents the mean of the distribution, while σ represents the standard deviation. The normal distribution has the following properties:

- **Symmetry:** The probability density function of the normal distribution is symmetrical around the mean μ.
- **Bell-shaped Curve:** The probability density function of the normal distribution has a bell-shaped curve, with the peak located at the mean μ.
- **Area Property:** The area under the probability density function between any two points around the mean μ equals the total probability within that interval.

## 2. Mathematical Principles of Normal Distribution Correlation Analysis

### 2.1 Covariance and Correlation Coefficient

#### 2.1.1 Definition and Calculation of Covariance

Covariance measures the strength of the linear relationship between two random variables. For two random variables X and Y, covariance is defined as:

Cov(X, Y) = E[(X - μ_X)(Y - μ_Y)]

Where E represents the expected value, and μ_X and μ_Y are the means of X and Y, respectively.

The formula for calculating covariance is:

Cov(X, Y) = (1/n) Σ[(x_i - μ_X)(y_i - μ_Y)]

Here, n is the sample size, and x_i and y_i represent the values of X and Y in the i-th sample, respectively.

#### 2.1.2 Definition and Calculation of Correlation Coefficient

The correlation coefficient is the normalized form of covariance, with values ranging from -1 to 1. It is defined as:

ρ(X, Y) = Cov(X, Y) / (σ_X σ_Y)

Where σ_X and σ_Y represent the standard deviations of X and Y, respectively.

The formula for calculating the correlation coefficient is:

ρ(X, Y) = (1/n) Σ[(x_i - μ_X)(y_i - μ_Y)] / (σ_X σ_Y)

The sign of the correlation coefficient indicates the direction of the linear relationship between X and Y:

* ρ(X, Y) > 0 indicates a positive correlation, meaning X and Y increase or decrease together.
* ρ(X, Y) < 0 indicates a negative correlation, meaning when X increases, Y decreases, and vice versa.
* ρ(X, Y) = 0 indicates no correlation, meaning there is no linear relationship between X and Y.

### 2.2 Statistical Inference of Correlation

#### 2.2.1 Hypothesis Testing for Correlation

Hypothesis testing for correlation is used to determine whether there is a significant linear relationship between two random variables. The hypothesis testing process is as follows:

1. **Formulate the null and alternative hypotheses:**
- Null hypothesis: H_0: ρ(X, Y) = 0 (X and Y are not correlated)
- Alternative hypothesis: H_1: ρ(X, Y) ≠ 0 (X and Y are correlated)

2. **Calculate the correlation coefficient and test statistic:**
- Calculate the sample correlation coefficient ρ(X, Y)
- Calculate the test statistic: t = ρ(X, Y) * √(n - 2) / √(1 - ρ(X, Y)^2)

3. **Determine the critical value:**
- Look up the critical value for a t-distribution table with a degree of freedom of n - 2

4. **Make a decision:**
- If |t| > t_α/2, then reject the null hypothesis, concluding that X and Y are correlated.
- Otherwise, accept the null hypothesis, concluding that X and Y are not correlated.

#### 2.2.2 Confidence Interval Estimation

Confidence interval estimation is used to estimate the true value of the correlation coefficient. The process is as follows:

1. **Calculate the correlation coefficient and standard error:**
- Calculate the sample correlation coefficient ρ(X, Y)
- Calculate the standard error: SE(ρ) = 1 / √(n - 2)

2. **Determine the confidence level:**
- Choose a confidence level, for example, 95%

3. **Calculate the confidence interval:**
- Calculate the confidence interval: ρ(X, Y) ± t_α/2 * SE(ρ)

Where t_α/2 is the two-tailed critical value for a degree of freedom of n - 2.

## 3. Practical Application of Normal Distribution Correlation Analysis in MATLAB

### 3.1 Data Import an