MATLAB Normal Distribution Confidence Intervals: Obtaining Reliable Estimates of Normal Distribution Parameters

# Introduction to Normal Distribution in MATLAB

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is widely present in nature and scientific research. Its probability density function is:

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

Where μ represents the mean of the normal distribution and σ represents the standard deviation.

The normal distribution has the following characteristics:

- **Symmetry:** The distribution curve is symmetric about the mean μ.
- **Bell-shaped curve:** The distribution curve is bell-shaped, with a maximum value at the mean μ.
- **Asymptotic behavior:** The distribution curve gradually approaches the horizontal line on both sides, meaning the probability density gradually decreases.

# Parameter Estimation of the Normal Distribution

### 2.1 Maximum Likelihood Estimation

#### 2.1.1 Construction of the Likelihood Function

The likelihood function is a function of the model parameters given the observed data. In the context of the normal distribution, the likelihood function is expressed as:

L(μ, σ^2 | x_1, x_2, ..., x_n) = (2πσ^2)^(-n/2) * exp(-∑(x_i - μ)^2 / (2σ^2))

Where:

* μ is the mean of the normal distribution
* σ^2 is the variance of the normal distribution
* x_1, x_2, ..., x_n are the observed data

#### 2.1.2 Solution Methods for Parameter Estimation

The maximum likelihood estimation (MLE) is a method for estimating model parameters by maximizing the likelihood function. For the normal distribution, MLE estimates can be analytically solved:

**Mean Estimation:**

μ_MLE = (1/n) * ∑x_i

**Variance Estimation:**

σ^2_MLE = (1/(n-1)) * ∑(x_i - μ_MLE)^2

### 2.2 Bayesian Estimation

#### 2.2.1 Selection of the Prior Distribution

Bayesian estimation is a method that combines prior distribution and observed data. The prior distribution represents prior beliefs about the parameters. For the normal distribution, the conjugate prior distribution is commonly chosen, which is:

* Mean prior: Normal distribution N(μ_0, σ_0^2)
* Variance prior: Inverse Gamma distribution Inv-Gamma(α, β)

#### 2.2.2 Derivation of the Posterior Distribution

By combining the prior distribution and the likelihood function, we can obtain the posterior distribution:

**Mean Posterior:**

μ_post | x_1, x_2, ..., x_n ~ N(μ_n, σ_n^2)

Where:

μ_n = (σ_0^2 * μ_0 + σ^2_MLE * μ_MLE) / (σ_0^2 + σ^2_MLE)
σ_n^2 = (σ_0^2 * σ^2_MLE) / (σ_0^2 + σ^2_MLE)

**Variance Posterior:**

σ^2_post | x_1, x_2, ..., x_n ~ Inv-Gamma(α + n/2, β + ∑(x_i - μ_MLE)^2 / 2)

# Theoretical Basis of Confidence Intervals for Normal Distribution

#### 3.1.1 Concepts of Confidence Level and Confidence Interval

**Confidence Level:**

The confidence level represents the degree of confidence in the accuracy of the estimated value. It is usually expressed as a percentage, such as 95% or 99%. The higher the confidence level, the greater the confidence in the estimated value.

**Confidence Interval:**

The confidence interval is an interval that contains the estimated value, with a specified confidence level probability. In other words, the confidence interval represents the probability that the estimated value falls within that interval.

#### 3.1.2 Formulas for Confidence Intervals of Normal Distribution

For the normal distribution, confidence intervals can be calculated using the following formula:

[μ - z * σ/√n, μ + z * σ/√n]

Where:

* μ is the mean of the normal distribution
* σ is the standard deviation of the normal distribution
* n is the sample size
* z is the z-score corresponding to the confidence level

For example, for a 95% confidence level, the z-score is 1.96.

### 3.2 Implementation of Confidence Intervals in MATLAB

#### 3.2.1 Normal Distribution Parameter Estimation Functions

MATLAB provides the `normfit` function to estimate the parameters of the normal distribution.