Laboratory - Pseudorandom Number Generation and Probability Distributions

Overview

This laboratory focuses on implementing and testing various random number generation methods and probability distributions using Java. The implementation includes a custom random number generator and methods for generating various probability distributions.

Tasks Overview

Lab_1:

Task 1: Linear Congruential Generator

Implementation of a basic pseudorandom number generator using the Linear Congruential Method (LCM).

Key Components:

• Parameters: $a=97, b=11, M=100, seed=1$
• Formula: $x_{n+1} = (a * x_n + b) mod M$
• Generator outputs values in ranges:
  • $[0, M)$ for integers
  • $[0, 1.0)$ for doubles
  • Custom ranges using linear transformation
  • Exponential distribution using inverse transform method

Pseudo-cycle Analysis

The implemented generator has a cycle length of 20 values:

8, 87, 50, 61, 28, 27, 30, 21, 48, 67, 10, 81, 68, 7, 90, 41, 88, 47, 70, 1, [repeat]

When normalized to [0,1) range:

0.08, 0.87, 0.50, 0.61, 0.28, 0.27, 0.30, 0.21, 0.48, 0.67, 
0.10, 0.81, 0.68, 0.07, 0.90, 0.41, 0.88, 0.47, 0.70, 0.01, [repeat]

Note: This relatively short cycle length indicates limitations of the chosen parameters for serious statistical applications.

Task 2: Custom Probability Distribution

Implementation of a specific probability distribution function with the following characteristics:

$ f(x) = { 2x + 1 for x ∈ [-1, 0] 4(x-0.5) for x ∈ [0, 2] } $

Key Features:

• Uses inverse transform method
• Expected value: 1/4
• Implemented in distribution_1() method
• Tested with 1000 iterations to verify mean

Task 3: Discrete Distribution

Implementation of a discrete probability distribution generator.

Implementation Details:

• Takes arrays of values (xx) and probabilities (p)
• Values: $[-2.0, -1.0, 0.0, 1.0, 2.0, 3.0, 4.0]$
• Probabilities: $[0.1, 0.2, 0.1, 0.05, 0.1, 0.3, 0.2]$
• Uses inverse transform method with cumulative probabilities
• Theoretical mean calculation included for verification

Technical Implementation Notes

RandomGenerator Class

Main class implementing all random generation methods:

• nextInt(): Basic integer generation
• nextDouble(): Generation of $[0,1)$ values
• nextDouble(low, high): Range transformation
• exponential(lambda): Exponential distribution
• distribution_1(): Custom distribution from Task 2
• discrete(xx, p): Discrete distribution

Testing Methods

Each task has its dedicated test method in Main class:

• testTask1(): Tests basic generator functionality
• testTask2(): Tests custom distribution
• testTask3(): Tests discrete distribution

Statistical Verification

• Empirical means calculated from generated samples
• Theoretical means calculated for comparison
• Absolute differences reported for accuracy assessment

Limitations and Considerations

1. The short cycle length $(20)$ of the basic generator limits its statistical applications
2. Parameters $(a, b, M)$ could be optimized for longer cycles
3. For serious applications, Java's built-in Random class would be more appropriate

Running the Tests

Execute the Main class to run all tests sequentially. Each test produces detailed output including:

• Generated values
• Empirical means
• Theoretical means
• Absolute differences between empirical and theoretical values