Program 1

Study of Random Numbers and Time Series

Introduction to numpy.random

import numpy as np
import matplotlib.pyplot as plt

# Generate a single random float number between 0 and 1
random_float = np.random.random()
print("Random float between 0 and 1:", random_float, "\n")

# Generate an array of 5 random floats between 0 and 1
random_floats_array = np.random.random(5)
print("Array of 5 random floats between 0 and 1:", random_floats_array, "\n")

# Generate a 2x3 array of random floats between 0 and 1
random_floats_2x3 = np.random.random((2, 3))
print("2x3 array of random floats between 0 and 1:", random_floats_2x3, "\n")

# Generate a random integer between 0 and 10 (inclusive)
random_integer = np.random.randint(0, 11)
print("Random integer between 0 and 10:", random_integer, "\n")

# Generate an array of 5 random integers between 0 and 10 (inclusive)
random_integers_array = np.random.randint(0, 11, size=5)
print("Array of 5 random integers between 0 and 10:", random_integers_array, "\n")

# Generate a 2x3 array of random integers between 0 and 10 (inclusive)
random_integers_2x3 = np.random.randint(0, 11, size=(2, 3))
print("2x3 array of random integers between 0 and 10:", random_integers_2x3, "\n")

# Generate a random float from a normal distribution with mean 0 and standard deviation 1
normal_float = np.random.randn()
print("Random float from normal distribution (mean 0, std 1):", normal_float, "\n")

# Generate an array of 5 random floats from a normal distribution with mean 0 and standard deviation 1
normal_floats_array = np.random.randn(5)
print("Array of 5 random floats from normal distribution (mean 0, std 1):", normal_floats_array, "\n")

# Generate a 2x3 array of random floats from a normal distribution with mean 0 and standard deviation 1
normal_floats_2x3 = np.random.randn(2, 3)
print("2x3 array of random floats from normal distribution (mean 0, std 1):", normal_floats_2x3, "\n")

# Generate random samples from a given list
sample_list = ['apple', 'banana', 'cherry']
random_sample = np.random.choice(sample_list)
print("Random sample from list:", random_sample, "\n")

# Generate 3 random samples from a given list with replacement
random_samples_with_replacement = np.random.choice(sample_list, size=3, replace=True)
print("3 random samples from list with replacement:", random_samples_with_replacement, "\n")

# Generate 3 random samples from a given list without replacement
random_samples_without_replacement = np.random.choice(sample_list, size=3, replace=False)
print("3 random samples from list without replacement:", random_samples_without_replacement, "\n")

# Shuffle a list in place
arr = np.array([1, 2, 3, 4, 5])
np.random.shuffle(arr)
shuffled_list = arr
print("Shuffled list:", shuffled_list, "\n")

# Generate a random float between a specified range (e.g., 5 to 10)
random_float_range = np.random.uniform(5, 10)
print("Random float between 5 and 10:", random_float_range, "\n")

# Generate an array of 5 random floats between a specified range (e.g., 5 to 10)
random_floats_range_array = np.random.uniform(5, 10, size=5)
print("Array of 5 random floats between 5 and 10:", random_floats_range_array, "\n")

# Generate a random integer from a Poisson distribution
poisson_integer = np.random.poisson(lam=5)
print("Random integer from Poisson distribution:", poisson_integer, "\n")

# Generate an array of 5 random integers from a Poisson distribution
poisson_integers_array = np.random.poisson(lam=5, size=5)
print("Array of 5 random integers from Poisson distribution:", poisson_integers_array, "\n")

# Generate random samples from a binomial distribution
binomial_sample = np.random.binomial(n=10, p=0.5)
print("Random sample from binomial distribution:", binomial_sample, "\n")

# Generate an array of 5 random samples from a binomial distribution
binomial_samples_array = np.random.binomial(n=10, p=0.5, size=5)
print("Array of 5 random samples from binomial distribution:", binomial_samples_array, "\n")

# Generate random numbers from a beta distribution
beta_sample = np.random.beta(a=2, b=5)
print("Random number from beta distribution:", beta_sample, "\n")

# Generate an array of 5 random numbers from a beta distribution
beta_samples_array = np.random.beta(a=2, b=5, size=5)
print("Array of 5 random numbers from beta distribution:", beta_samples_array, "\n")

# Generate random numbers from a gamma distribution
gamma_sample = np.random.gamma(shape=2, scale=2)
print("Random number from gamma distribution:", gamma_sample, "\n")

# Generate an array of 5 random numbers from a gamma distribution
gamma_samples_array = np.random.gamma(shape=2, scale=2, size=5)
print("Array of 5 random numbers from gamma distribution:", gamma_samples_array, "\n")

# Generate random samples from a chi-square distribution
chi_square_sample = np.random.chisquare(df=2)
print("Random sample from chi-square distribution:", chi_square_sample, "\n")

# Generate an array of 5 random samples from a chi-square distribution
chi_square_samples_array = np.random.chisquare(df=2, size=5)
print("Array of 5 random samples from chi-square distribution:", chi_square_samples_array, "\n")

# Generate random numbers from an exponential distribution
exponential_sample = np.random.exponential(scale=1)
print("Random number from exponential distribution:", exponential_sample, "\n")

# Generate an array of 5 random numbers from an exponential distribution
exponential_samples_array = np.random.exponential(scale=1, size=5)
print("Array of 5 random numbers from exponential distribution:", exponential_samples_array, "\n")

# Generate a random float from a log-normal distribution
lognormal_sample = np.random.lognormal(mean=0, sigma=1)
print("Random float from log-normal distribution:", lognormal_sample, "\n")

# Generate an array of 5 random floats from a log-normal distribution
lognormal_samples_array = np.random.lognormal(mean=0, sigma=1, size=5)
print("Array of 5 random floats from log-normal distribution:", lognormal_samples_array, "\n")

# Generate random samples from a standard normal distribution and compute their mean and variance
normal_samples = np.random.randn(1000)
mean_normal_samples = np.mean(normal_samples)
variance_normal_samples = np.var(normal_samples)
print(f"Mean of normal samples: {mean_normal_samples}, Variance of normal samples: {variance_normal_samples}", "\n")

# Generate random integers from a specified range (inclusive) and size
random_integers_range_size = np.random.randint(low=1, high=10, size=(3, 4))
print("Random integers from specified range and size:", random_integers_range_size, "\n")

# Generate random floats from a uniform distribution in the range [0, 1) with size
uniform_floats_array = np.random.random_sample(size=(2, 2))
print("Array of random floats from uniform distribution [0, 1):", uniform_floats_array, "\n")

# Generate a random permutation of a given sequence
random_permutation = np.random.permutation([1, 2, 3, 4, 5])
print("Random permutation of sequence:", random_permutation, "\n")

# Generate random numbers from a multivariate normal distribution
mean = [0, 0]
cov = [[1, 0.5], [0.5, 1]]  # covariance matrix
multivariate_normal_samples = np.random.multivariate_normal(mean, cov, size=3)
print("Random numbers from multivariate normal distribution:", multivariate_normal_samples, "\n")

# Generate random numbers from a Dirichlet distribution
alpha = [1.0, 1.0, 1.0]
dirichlet_samples = np.random.dirichlet(alpha)
print("Random numbers from Dirichlet distribution:", dirichlet_samples, "\n")

# Generate random numbers from a Weibull distribution
weibull_samples = np.random.weibull(a=1.5, size=5)
print("Random numbers from Weibull distribution:", weibull_samples, "\n")

# Generate random integers from a binomial distribution with varying number of trials
binomial_integer_varying_trials = np.random.binomial(n=20, p=0.3)
print("Random integer from binomial distribution with varying trials:", binomial_integer_varying_trials, "\n")

# Generate a random float from a standard Cauchy distribution
cauchy_float = np.random.standard_cauchy()
print("Random float from standard Cauchy distribution:", cauchy_float, "\n")

# Generate an array of random floats from a standard Cauchy distribution
cauchy_floats_array = np.random.standard_cauchy(size=5)
print("Array of random floats from standard Cauchy distribution:", cauchy_floats_array, "\n")

# Generate random samples from a log-normal distribution
lognormal_samples = np.random.lognormal(mean=0, sigma=1, size=5)
print("Random samples from log-normal distribution:", lognormal_samples, "\n")

# Generate random samples from a geometric distribution
geometric_sample = np.random.geometric(p=0.5)
print("Random sample from geometric distribution:", geometric_sample, "\n")

# Generate an array of random samples from a geometric distribution
geometric_samples_array = np.random.geometric(p=0.5, size=5)
print("Array of random samples from geometric distribution:", geometric_samples_array, "\n")

# Generate random integers from a multinomial distribution
multinomial_samples = np.random.multinomial(n=10, pvals=[0.2, 0.3, 0.5])
print("Random integers from multinomial distribution:", multinomial_samples, "\n")

# Generate random samples from a beta distribution and compute statistics
beta_samples = np.random.beta(a=2, b=5, size=1000)
mean_beta_samples = np.mean(beta_samples)
variance_beta_samples = np.var(beta_samples)
print(f"Mean of beta samples: {mean_beta_samples}, Variance of beta samples: {variance_beta_samples}", "\n")

# Generate random samples from a normal distribution and plot a histogram
normal_samples_for_histogram = np.random.randn(1000)
plt.hist(normal_samples_for_histogram, bins=30, density=True)
plt.title('Histogram of Normally Distributed Samples')
plt.show()

# Generate a 3x4 array of random floats from a normal distribution with a different mean and variance
normal_floats_3x4 = np.random.normal(loc=10, scale=5, size=(3, 4))
print("3x4 array of random floats from normal distribution (mean 10, std 5):", normal_floats_3x4, "\n")

# Generate random samples from a Pareto distribution
pareto_samples = np.random.pareto(a=5, size=5)
print("Random samples from Pareto distribution:", pareto_samples, "\n")

# Generate random samples from a standard normal distribution and compute their quantiles
normal_samples_for_quantiles = np.random.randn(1000)
percentile_25 = np.percentile(normal_samples_for_quantiles, 25)
percentile_75 = np.percentile(normal_samples_for_quantiles, 75)
print(f"25th percentile of normal samples: {percentile_25}")
print(f"75th percentile of normal samples: {percentile_75}", "\n")

# Generate random numbers from a normal distribution with specified mean and variance
mean_normal = 5
std_dev_normal = 2
normal_samples_custom = np.random.normal(loc=mean_normal, scale=std_dev_normal, size=5)
print(f"Random numbers from normal distribution (mean {mean_normal}, std {std_dev_normal}):", normal_samples_custom, "\n")

# Generate random samples from a uniform distribution over a specific interval
uniform_samples_interval = np.random.uniform(low=10, high=20, size=5)
print("Random samples from uniform distribution (10 to 20):", uniform_samples_interval, "\n")

Output 1

Program 2

Study of Random Numbers and Time Series

Generation of uniform random number between (0,1) using LCG method

Output 2

Program 3

Study of Random Numbers and Time Series

Create a random variate on the interval (a, b)

Output 3

Program 4

Study of Random Numbers and Time Series

plot the autocorrelation of a random series

Output 4

Program 5

Study of Random Numbers and Time Series

statistical parameters such as mean, median, mode, and standard deviation for a random time series

Output 5

Application in Statistical Mechanics

Study of Random Numbers and Time Series

Study of Random Numbers and Time Series

Study of Random Numbers and Time Series

Study of Random Numbers and Time Series

Study of Random Numbers and Time Series