Sampling from Probability Distributions: Monte Carlo Simulation as a Powerful Tool in MATLAB

# Probability Distribution Sampling: A Powerful Tool for Monte Carlo Simulation in MATLAB

## 1. Overview of Probability Distribution Sampling

Probability distribution sampling is a method that estimates the parameters of probability distributions or performs probabilistic calculations by randomly generating sample data. It has broad applications in fields such as statistics, machine learning, and financial modeling.

Probability distribution sampling is mainly divided into two categories:

- **Uniform Sampling:** Randomly generating sample data within a given interval.
- **Non-uniform Sampling:** Generating sample data according to a given probability distribution function, so that the distribution of the sample data matches the given probability distribution.

## 2. The Basic Principles of Monte Carlo Simulation

### 2.1 Random Number Generation and Probability Distribution

The core of Monte Carlo simulation is random number generation, which relies on pseudo-random number generators (PRNGs) to produce sequences of numbers that appear random. These PRNGs use deterministic algorithms, but through clever design, they can produce sequences with statistical properties similar to those of truly random sequences.

In Monte Carlo simulation, random numbers are used to simulate probability distribut***mon probability distributions include:

- **Uniform Distribution:** Each value within a specified range has an equal probability of occurrence.
- **Normal Distribution:** Also known as the bell curve, it describes a distribution where most values are concentrated around the mean.
- **Exponential Distribution:** It describes how the frequency of events decreases exponentially over time.
- **Poisson Distribution:** It describes the number of events that occur within a given time interval.

### 2.2 Principles and Methods of Monte Carlo Integration

Monte Carlo integration is a special application of Monte Carlo simulation used to estimate integrals. Essentially, integration is the process of finding the area under a curve. Monte Carlo integration estimates an integral by generating random points and calculating the area under the curve for these points.

Specifically, the steps of Monte Carlo integration are as follows:

1. **Generate Random Points:** Generate a set of random points according to a probability distribution.
2. **Calculate Point Values:** For each random point, calculate its value under the curve.
3. **Estimate the Integral:** Estimate the integral by scaling the average value of all points.

import random

def monte_carlo_integral(f, a, b, n):
   """
   Estimates the integral of function f over the interval [a, b] using Monte Carlo integration.

   Parameters:
     f: The function to integrate.
     a: The lower limit of integration.
     b: The upper limit of integration.
     n: The number of random points.

   Returns:
     The estimated value of the integral.
   """

   # Generate random points
   points = [random.uniform(a, b) for _ in range(n)]

   # Calculate point values
   values = [f(x) for x in points]

   # Estimate the integral
   integral = (b - a) * sum(values) / n

   return integral

The advantage of Monte Carlo integration is that it can be applied to complex functions that may not be integrable analytically. Additionally, it can be parallelized, making it highly efficient when dealing with large datasets.

## 3.1 Random Number Generation Functions in MATLAB

MATLAB provides various random number generation functions for generating random numbers that conform to different probability distributions. These functions include:

- `rand`: Generates uniformly distributed random numbers, with a range of [0, 1).
- `randn`: Generates normally distributed random numbers, with a mean of 0 and a standard deviation of 1.
- `randsample`: Randomly samples a specified number of elements from a given set.
- `unifrnd`: Generates uniformly distributed random numbers, with a specified range of [a, b).
- `normrnd`: Generates normally distributed random numbers, with specified mean and standard deviation.

**Code Block:**

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

% Generate normally distributed random numbers
normNum = randn(1, 10);

% Randomly sample 5 elements from the set 1:10
sampleNum = randsample(1:10, 5);

% Generate uniformly distributed random numbe