### Illustration of the linear congruential method.  
### Using Python code to generate random numbers.

# develop the algorithm to run the generator.

# Python code

a = 5
c = 7
m = 8

n = 20	# number of random values to be generated

x = [0]*n
print(x)

x[0] = 4		# seed or x_0

for i in range(1,n):
  	x[i] = (a*x[i-1] + c) 
  	x[i] = x[i] % m		# this calculates the remainder, i.e., calculates the mod

x = x/m 

print(x)

# Question:  What does m control in this random number generator?  Why should the value of m be large?

#############################################################################################################

# A function that generates psudo random uniform values using the linear congruential method.

import numpy as np

def RANU(seed,n):
  # a,c,m are the values need for the linear congruential method.
	# seed is the value of the seed for the random number generator.
	# n is the length of the vector returned.
	
	a = 65539
	c = 0
	m = 2**31
	
	x = np.zeros(n)		# creates the output vecotor of length n
	x[0] = seed
	for i in range(1,n):
		x[i] = (a*x[i-1] + c)   # this calculates the remainder, i.e., calculates the mod
		x[i] = x[i] % m		 # dividing by m gives values between [0,1)
	x = x/m
	return x
	
# A function that generates psudo random exponential values using the inverse cdf method.

def RANE(seed,n):
  # seed is the seed for the random number generator.
	# n is the length of the vector returned.
	
	x = RANU(seed,n)
	y = -np.log(x)
	return y

### Illustration of the RANU.

seed = 4
n = 20

x = RANU(seed,n)
print(x)

# make a scatter plot of pairs of values.

import numpy as np
import matplotlib.pyplot as plt

# Number of pairs
n = 100

# Generate n random uniform values between 0 and 1
x_values = np.random.uniform(0, 1, n)
y_values = np.random.uniform(0, 1, n)

# Create sequential pairs
pairs = [(x_values[i], y_values[i]) for i in range(n)]

# Extract x and y coordinates from pairs
x_coords = [pair[0] for pair in pairs]
y_coords = [pair[1] for pair in pairs]

# Create a scatter plot
plt.figure(figsize=(8, 6))
plt.scatter(x_coords, y_coords, color='blue', alpha=0.5)
plt.xlabel('X Values')
plt.ylabel('Y Values')
plt.title('Scatter Plot of Sequential Pairs of Random Uniform Values')
plt.grid(True)

# Show the plot
plt.show()

## Illustration of the RANE.

seed = 4
n = 20

x = RANE(seed,n)
print(x)

# make a scatter plot of pairs of values.

import numpy as np
import matplotlib.pyplot as plt

# Number of pairs
n = 100

# Generate n random values from an exponential distribution with lambda = 1 (default rate parameter)
x_values = np.random.exponential(scale=1, size=n)
y_values = np.random.exponential(scale=1, size=n)

# Create sequential pairs
pairs = [(x_values[i], y_values[i]) for i in range(n)]

# Extract x and y coordinates from pairs
x_coords = [pair[0] for pair in pairs]
y_coords = [pair[1] for pair in pairs]

# Create a scatter plot
plt.figure(figsize=(8, 6))
plt.scatter(x_coords, y_coords, color='blue', alpha=0.5)
plt.xlabel('X Values')
plt.ylabel('Y Values')
plt.title('Scatter Plot of Sequential Pairs of Exponential Random Variables')
plt.grid(True)

# Show the plot
plt.show()