【Advanced】MATLAB Statistics Toolbox: Statistics Toolbox User Guide

# 2.1 Types of Probability Distributions and Parameter Estimation

Probability distributions describe the probability of possible outcomes of random variables. MATLAB's Statistics Toolbox provides a variety of probability distribution functions, including normal, t, and chi-square distributions.

### 2.1.1 Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a symmetric bell-shaped distribution. Its probability density function is:

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

where:

- x is the random variable
- μ is the mean
- σ is the standard deviation

### 2.1.2 t Distribution

The t distribution is a generalization of the normal distribution and is used for smaller sample sizes. Its probability density function is:

f(x) = (Γ((ν + 1) / 2) / (Γ(ν / 2) * √(πν))) * (1 + (x² / ν))^(-(ν + 1) / 2)

where:

- x is the random variable
- ν is the degrees of freedom

### 2.1.3 Chi-Square Distribution

The chi-square distribution is a non-negative distribution used to assess goodness of fit of data. Its probability density function is:

f(x) = (1 / (2^(ν / 2) * Γ(ν / 2))) * x^(ν / 2 - 1) * exp(-x / 2)

where:

- x is the random variable
- ν is the degrees of freedom

# 2. Probability Distributions and Statistical Inference

### 2.1 Types of Probability Distributions and Parameter Estimation

Probability distributions describe the probability of possible outcomes of random variables. MATLAB's Statistics Toolbox offers a variety of probability distribution functions for fitting data and estimating their parameters.

#### 2.1.1 Normal Distribution

The normal distribution, also known as the Gaussian distribution, is one of the most commonly used distributions in statistics. It features a symmetrical bell-shaped curve defined by two parameters: the mean and the standard deviation.

% Generate normal distribution data
data = normrnd(0, 1, 1000);

% Estimate normal distribution parameters
params = normfit(data);
mean_est = params(1);
std_est = params(2);

% Plot the normal distribution curve
figure;
histogram(data, 50);
hold on;
x = linspace(-5, 5, 100);
y = normpdf(x, mean_est, std_est);
plot(x, y, 'r', 'LineWidth', 2);
legend('Data', 'Fitted Normal Distribution');
title('Normal Distribution Fit');
xlabel('Value');
ylabel('Frequency');

#### 2.1.2 t Distribution

The t distribution is an extension of the normal distribution, used when sample sizes are small. It shares a similar shape with the normal distribution but has heavier tails.

% Generate t distribution data
data = trnd(5, 1000);

% Estimate t distribution parameters
params = tfit(data);
mu_est = params(1);
sigma_est = params(2);
nu_est = params(3);

% Plot t distribution curve
figure;
histogram(data, 50);
hold on;
x = linspace(-5, 5, 100);
y = tpdf(x, nu_est, mu_est, sigma_est);
plot(x, y, 'r', 'LineWidth', 2);
legend('Data', 'Fitted t Distribution');
title('t Distribution Fit');
xlabel('Value');
ylabel('Frequency');

#### 2.1.3 Chi-Square Distribution

The chi-square distribution is a non-negative distribution used to test the difference between two probability distributions. It has a skewed shape and is defined by the degrees of freedom parameter.

% Generate chi-square distribution data
data = chi2rnd(5, 1000);

% Estimate chi-square distribut