> For the complete documentation index, see [llms.txt](https://suyashs-organization-3.gitbook.io/csitlabs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://suyashs-organization-3.gitbook.io/csitlabs/5th_semester/simulation_and_modeling/1_random_number_generation.md).

# Lab 1: Random Number Generation

## Random Number

A random number is a number that is generated by a process, whose outcome is unpredictable, and which cannot be subsequently reliably reproduced. Random numbers are important in statistical analysis and probability theory.

## Pseudo Random Number

A pseudo-random number generator (PRNG) is a finite state machine with an initial value called the seed \[4]. Upon each request, a pseudo-random number generator will return a pseudo-random number based on some deterministic computation or algorithm. The generation of pseudo-random numbers is an important and common task in computer programming. The word pseudo-random is used because the numbers are not random in the sense that they are completely unpredictable and independent of each other. However, they are sufficiently random for practical purposes.

## Methods of Generating Random Numbers

1. Linear Congruential Method
   1. Mix Linear Congruential Method
   2. Additive Congruential Method
   3. Multiplicative Congruential Method
   4. Arithmetic Congruential Method
2. Mid Square Method

## Linear Congruential Method

1. ### Mix Linear Congruential method:

#### Definition

Linear congruential generators (LCGs) are a class of pseudorandom number generators (PRNGs) that generate a sequence of integers \[0, m-1] calculated with a linear equation of the form:

$$
Xn+1 = (aXn + c) mod m
$$

where X is the sequence of pseudorandom values m, 0 < m - the "modulus" a, 0 < a < m - the "multiplier" c, 0 <= c < m - the "increment" X0, 0 <= X0 < m - the "seed" or "start value".

#### Algorithm

1. Start
2. Input the values of a, c, m and X0
3. Set i = 0
4. Generate `Xn+1 = (aXn + c) mod m`
5. Set i = i + 1
6. If i < n, goto step 4, otherwise goto step 7
7. End

#### Program

[Mixed Linear Congruential Method](https://github.com/sthsuyash/CSIT_Labs/blob/main/5th_Semester/Simulation_and_Modeling/1_Random_number_Generation/a_mixed_congruential.cpp)

#### Output

Generate by running the program.

2. ### Multiplicative Congruential Method:

#### Definition

Multiplicative congruential generators (MCGs) are a class of pseudorandom number generators (PRNGs) that generate a sequence of integers \[0, m-1] calculated with a linear equation of the form:

$$
Xn+1 = (aXn) mod m
$$

where X is the sequence of pseudorandom values m, 0 < m - the "modulus" a, 0 < a < m - the "multiplier" X0, 0 <= X0 < m - the "seed" or "start value".

#### Algorithm

1. Start
2. Input the values of a, m and X0
3. Set i = 0
4. Generate `Xn+1 = (aXn) mod m`
5. Set i = i + 1
6. If i < n, goto step 4, otherwise goto step 7.
7. End

#### Program

[Multiplicative Congruential Method](https://github.com/sthsuyash/CSIT_Labs/blob/main/5th_Semester/Simulation_and_Modeling/1_Random_number_Generation/b_linear_additive_congruential.cpp)

#### Output

Generate by running the program.

3. ### Additive Congruential Method:

#### Definition

Additive congruential generators (ACGs) are a class of pseudorandom number generators (PRNGs) that generate a sequence of integers \[0, m-1] calculated with a linear equation of the form:

$$
Xn+1 = (Xn + c) mod m
$$

where X is the sequence of pseudorandom values m, 0 < m - the "modulus" c, 0 <= c < m - the "increment" X0, 0 <= X0 < m - the "seed" or "start value".

#### Algorithm

1. Start
2. Initialize X1 and X2 as seed values.
3. Input n (number of random values to generate) and m (modulus value).
4. Display "Generated numbers: X1 X2".
5. Set i = 2.
6. Repeat the following steps n - 2 times:
   1. Calculate the next random number:

      next = (X1 + X2) % m
   2. Display next.
   3. Update X1 and X2:

      X1 = X2 X2 = next
   4. Increment i by 1.
7. End the loop.
8. Display "Random numbers generated."
9. End

#### Program

[Additive Congruential Method](https://github.com/sthsuyash/CSIT_Labs/blob/main/5th_Semester/Simulation_and_Modeling/1_Random_number_Generation/c_linear_multiplicative_congruential.cpp)

#### Output

Generate by running the program.

4. ### Arithmetic Congruential Method:

#### Definition

Arithmetic congruential generators (ACGs) are a class of pseudorandom number generators (PRNGs) that generate a sequence of integers \[0, m-1] calculated with a linear equation of the form:

$$
Xn+1 = Xn-1 + Xn
$$

where X is the sequence of pseudorandom values m, 0 < m - the "modulus" X0, 0 <= X0 < m - the "seed" or "start value".

#### Algorithm

1. Start
2. Initialize X1 and X2 as seed values.
3. Input n (number of random values to generate) and m (modulus value).
4. Display "Generated numbers: X1 X2".
5. Set i = 2.
6. Repeat the following steps n - 2 times:
   1. Calculate the next random number:

      next = (X1 + X2) % m
   2. Display next.
   3. Update X1 and X2:

      X1 = X2 X2 = next
   4. Increment i by 1.
7. End the loop.
8. Display "Random numbers generated."
9. End

#### Program

[Arithmetic Congruential Method](https://github.com/sthsuyash/CSIT_Labs/blob/main/5th_Semester/Simulation_and_Modeling/1_Random_number_Generation/d_linear_arithmetic_congruential.cpp)

#### Output

Generate by running the program.

## Mid Square Method

#### Definition

The middle-square method is a method of generating pseudorandom numbers. In practice it is not a good method, since its period is usually very short and it has some severe weaknesses, such as the output sequence almost always converging to zero.

#### Algorithm

1. Start
2. Input the value of seed X0
3. Set i = 0
4. Generate `Xn+1 = (Xn)^2`
5. Set i = i + 1
6. If i < n, goto step 4, otherwise goto step 7.
7. End

#### Program

[Middle Square Method](https://github.com/sthsuyash/CSIT_Labs/blob/main/5th_Semester/Simulation_and_Modeling/1_Random_number_Generation/e_mid_square.cpp)

#### Output

Generate by running the program.