Bayesian Analysis of Normal Distribution in MATLAB: Exploring Normal Distribution from a Bayesian Perspective

# 1. Normal Distribution Basics

The normal distribution, also known as the Gaussian distribution, is a common probability distribution widely used in nature and the social sciences. Its probability density function is given by:

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

Where μ represents the mean and σ the standard deviation. The normal distribution has the following characteristics:

- **Symmetry:** It is symmetrically distributed about the mean.
- **Bell Curve:** It has a bell-shaped curve, highest in the middle and gradually decreases on both sides.
- **68-95-99.7 Rule:** Approximately 68%, 95%, and 99.7% of the data lie within one, two, and three standard deviations (σ) of the mean (μ), respectively.

# 2. Principles of Bayesian Analysis

### 2.1 Bayes' Theorem and Prior Distribution

Bayesian analysis is a statistical method based on Bayes' theorem that combines prior knowledge with observed data to update probability distributions. Bayes' theorem describes the probability of an event occurring given some known conditions, and its formula is:

P(A | B) = P(B | A) * P(A) / P(B)

Where:

- P(A | B) is the probability of event A occurring given event B has occurred (posterior probability)
- P(B | A) is the probability of event B occurring given event A has occurred (likelihood function)
- P(A) is the prior probability of event A occurring
- P(B) is the probability of event B occurring

In Bayesian analysis, the prior distribution represents beliefs about parameters before observing the data. It can be any probability distribution, but typically a distribution that matches the expected parameter values is chosen.

### 2.2 Calculation of the Posterior Distribution

The posterior distribution is obtained by combining the prior distribution with the likelihood function, representing updated beliefs about parameters after observing the data. The calculation method for the posterior distribution is:

P(θ | x) = P(x | θ) * P(θ) / P(x)

Where:

- P(θ | x) is the posterior probability of parameter θ given observed data x
- P(x | θ) is the likelihood of observing data x given parameter θ
- P(θ) is the prior probability of parameter θ
- P(x) is the marginal distribution of observed data x (normalization constant)

The calculation of the posterior distribution often involves complex integrals, and thus approximate methods such as Monte Carlo simulation or variational inference are commonly used.

# 3. Bayesian Analysis of Normal Distribution in MATLAB

### 3.1 MATLAB Functions for Normal Distribution

MATLAB offers a comprehensive library of functions for handling normal distributions, including:

| Function | Purpose |
|---|---|
| `normpdf` | Computes the probability density function of a normal distribution |
| `normcdf` | Computes the cumulative distribution function of a normal distribution |
| `norminv` | Computes the inverse cumulative distribution function of a normal distribution |
| `normrnd` | Generates random numbers from a normal distribution |
| `normfit` | Fits a normal distribution to data |

**Example Code:**

% Define parameters for the normal distribution
mu = 0;   % Mean
sigma = 1;   % Standard deviation

% Compute the probability density function
x = -3:0.1:3;
y = normpdf(x, mu, sigma);

% Plot the probability density function
figure;
plot(x, y);
xlabel('x');
ylabel('Probability Density');
title('Normal Distribution Probability Density Function');

**Logical Analysis:**

* The `normpdf` function takes three parameters: x (input value), mu (mean), and sigma (standard deviation).
* The `plot` function dr