【Fundamentals】In-depth Exploration of Random Number Generation and Statistical Analysis in MATLAB

**1. Random Number Generation in MATLAB**

MATLAB offers a variety of pseudo-random number generators (PRNGs) that produce deterministic sequences of numbers that appear to be random. The most commonly used PRNG is the `rand` function, which generates uniformly distributed random numbers in the range [0,1].

% Generate 10 uniformly distributed random numbers
rand_numbers = rand(1, 10);

MATLAB also provides functions for generating random numbers with other distributions, such as `randn` (normal distribution), `randperm` (permutation), and `poissrnd` (Poisson distribution).

**2. Practical Random Number Generation**

### 2.1 Generation of Random Variables

#### 2.1.1 Discrete Random Variables

**Generating Binomial Distributed Random Variables**

n = 10;   % Number of trials
p = 0.5;   % Probability of success
x = binornd(n, p, 1000);   % Generate 1000 binomially distributed random variables

**Logical Analysis:**

The `binornd` function is used to generate binomially distributed random variables. It accepts three parameters:

* `n`: Number of trials
* `p`: Probability of success
* `size`: Number of random variables to generate

The function returns a vector of random variables of the specified size.

**Generating Poisson Distributed Random Variables**

lambda = 5;   % Average rate of occurrence
x = poissrnd(lambda, 1000);   % Generate 1000 Poisson distributed random variables

**Logical Analysis:**

The `poissrnd` function is used to generate Poisson distributed random variables. It accepts two parameters:

* `lambda`: Average rate of occurrence
* `size`: Number of random variables to generate

The function returns a vector of random variables of the specified size.

#### 2.1.2 Continuous Random Variables

**Generating Uniformly Distributed Random Variables**

a = 0;   % Lower bound
b = 10;   % Upper bound
x = unifrnd(a, b, 1000);   % Generate 1000 uniformly distributed random variables

**Logical Analysis:**

The `unifrnd` function is used to generate uniformly distributed random variables. It accepts three parameters:

* `a`: Lower bound
* `b`: Upper bound
* `size`: Number of random variables to generate

The function returns a vector of random variables of the specified size.

**Generating Normally Distributed Random Variables**

mu = 0;   % Mean
sigma = 1;   % Standard deviation
x = normrnd(mu, sigma, 1000);   % Generate 1000 normally distributed random variables

**Logical Analysis:**

The `normrnd` function is used to generate normally distributed random variables. It accepts three parameters:

* `mu`: Mean
* `sigma`: Standard deviation
* `size`: Number of random variables to generate

The function returns a vector of random variables of the specified size.

### 2.2 Generation of Random Sequences

#### 2.2.1 White Noise

**Generating White Noise Sequence**

N = 1000;   % Sequence length
x = randn(1, N);   % Generate 1000 white noise samples

**Logical Analysis:**

The `randn` function is used to generate normally distributed random numbers. It accepts one parameter:

* `size`: Dimensionality of the generated random numbers

The function returns a matrix of random numbers of the specified size.

#### 2.2.2 Brownian Motion

**Generating Brownian Motion Sequence**

N = 1000;   % Sequence length
dt = 0.01;   % Time step
sigma = 0.1;   % Standard deviation

x = zeros(1, N);
for i = 2:N
     x(i) = x(i-1) + sqrt(dt) * normrnd(0, sigma);
end

**Logical Analysis:**

Brownian motion is a stochastic process where the increments are normally distributed. This code uses the Euler-Maruyama method to generate the sequence of Brownian motion.

* `zeros(1, N)` creates a zero vector of length `N`.
* The loop starts at `i = 2` since the initial value of Brownian motion is usually zero.
* In each iteration, `x(i)` is updated to the previous value `x(i-1)` plus the square root of the time step `dt` multiplied by a normally distributed random number `normrnd(0, sigma)`.

The code generates a Brownian motion sequence of length `N`, stored in the vector `x`.

**3. Descriptive Statistics**

#### 3.1 Me***

***mon measures of central tendency include:

***Mean (Average):** The sum of all data values di