Application in Statistical Mechanics - Study of Random Numbers and Time Series

Program 1

Study of Random Numbers and Time Series

Python 3.11 Ready

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")