Visualizing Statistical Distributions with Python

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm, binom, poisson, expon, uniform, bernoulli, chi2, t

# Normal Distribution
mu = 0  # Mean
sigma = 1  # Standard deviation
x = np.linspace(-5, 5, 1000)
plt.plot(x, norm.pdf(x, mu, sigma), label='Normal Distribution')
plt.title('Normal Distribution')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.legend()
plt.show()

# Exponential Distribution
lam = 1  # Decay rate
x = np.linspace(0, 5, 1000)
plt.plot(x, expon.pdf(x, scale=1/lam), label='Exponential Distribution')
plt.title('Exponential Distribution')
plt.xlabel('Time')
plt.ylabel('Probability Density')
plt.legend()
plt.show()

# Parameter of the Bernoulli distribution
p = 0.4  # Probability of success (1)

# Possible values of the Bernoulli random variable
x = [0, 1]

# Calculation of probability mass function
pmf_values = bernoulli.pmf(x, p)

# Creating the plot
bar_width = 0.3
x_pos = np.array([0, 0.6])  # Adjust these values to change the spacing
plt.bar(x_pos, pmf_values, width=bar_width, color='blue', alpha=0.7, label='Bernoulli Distribution')

# Setting labels and title
plt.title(f'Bernoulli Distribution (p = {p:.2f})')
plt.xlabel('Value')
plt.ylabel('Probability Mass')
plt.xticks(x_pos, ['0', '1'])  # Set x-ticks at bar positions
plt.legend()
plt.grid(True, axis='y', linestyle='--', alpha=0.7)  # Adds horizontal grid to improve readability

# Set x-axis limits to focus on the bars
plt.xlim(-0.2, 0.8)
plt.show()

# Binomial Distribution
n = 4  # Number of trials
p = 0.5  # Probability of success
x = np.arange(0, n+1)

# Calculate PMF
pmf_values = binom.pmf(x, n, p)

# Create the plot
bar_width = 0.8
plt.bar(x, pmf_values, width=bar_width, color='blue', alpha=0.7, label='Binomial Distribution')

# Set labels and title
plt.title(f'Binomial Distribution (n={n}, p={p})')
plt.xlabel('Number of Successes')
plt.ylabel('Probability')

# Set x-ticks to integers
plt.xticks(x)

# Add legend and grid
plt.legend()
plt.grid(True, axis='y', linestyle='--', alpha=0.7)

# Adjust x-axis limits for better appearance
plt.xlim(-0.5, n+0.5)
plt.show()

# Poisson Distribution
lam = 5  # Rate or mean number of events
x = np.arange(0, 20)

# Calculate PMF
pmf_values = poisson.pmf(x, lam)

# Create the plot
bar_width = 0.8
plt.bar(x, pmf_values, width=bar_width, color='blue', alpha=0.7, label='Poisson Distribution')

# Set labels and title
plt.title(f'Poisson Distribution (λ = {lam})')
plt.xlabel('Number of Events')
plt.ylabel('Probability')

# Set x-ticks
plt.xticks(np.arange(0, 20, 2))  # Set x-ticks every 2 units for better readability

# Add legend and grid
plt.legend()
plt.grid(True, axis='y', linestyle='--', alpha=0.7)

# Adjust x-axis limits for better appearance
plt.xlim(-0.5, 19.5)
plt.show()

# Uniform Distribution
a = 0  # Lower bound
b = 10  # Upper bound

# Generate x values
x = np.linspace(a-1, b+1, 1000)

# Calculate PDF
pdf_values = uniform.pdf(x, loc=a, scale=b-a)

# Create the plot
plt.figure(figsize=(10, 6))
plt.plot(x, pdf_values, color='blue', linewidth=2, label='Uniform Distribution')

# Fill the area under the curve within the bounds
plt.fill_between(x, pdf_values, where=((x >= a) & (x <= b)), color='blue', alpha=0.3)

# Set labels and title
plt.title(f'Uniform Distribution (a={a}, b={b})')
plt.xlabel('Value')
plt.ylabel('Probability Density')

# Add legend and grid
plt.legend()
plt.grid(True, linestyle='--', alpha=0.7)

# Set axis limits
plt.xlim(a-1, b+1)
plt.ylim(0, uniform.pdf(a, loc=a, scale=b-a) * 1.1)

# Add vertical lines at bounds
plt.axvline(x=a, color='gray', linestyle='--')
plt.axvline(x=b, color='gray', linestyle='--')
plt.show()

# Set range for degrees of freedom
degrees_of_freedom = range(1, 11)

# Create a range of x values for plotting
x = np.linspace(0, 20, 1000)

# Plot chi-squared distributions for each degree of freedom
plt.figure(figsize=(10, 6))
for k in degrees_of_freedom:
    plt.plot(x, chi2.pdf(x, k), label=f'df = {k}')

plt.title('Chi-Squared Distributions for Degrees of Freedom from 1 to 10')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.legend(title='Degrees of Freedom')
plt.grid(True)
plt.show()

# Set up a range for x values to cover enough area for both distributions
x_range = np.linspace(-5, 5, 1000)

# Compute the probability density functions for a t-distribution with 10 degrees of freedom and a normal distribution
t_distribution = t.pdf(x_range, df=10)
normal_distribution = norm.pdf(x_range)

# Plot both distributions for comparison
plt.figure(figsize=(10, 6))
plt.plot(x_range, t_distribution, label='Student\\'s t-distribution, df=10')
plt.plot(x_range, normal_distribution, label='Normal distribution')
plt.title('Comparison of Student\\'s t-Distribution and Normal Distribution')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.legend()
plt.grid(True)
plt.show()

# Define the sigmoid function
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

# Set up a range for x values to display the sigmoid curve
x_values = np.linspace(-10, 10, 400)

# Compute the sigmoid function for these x values
sigmoid_values = sigmoid(x_values)

# Plot the sigmoid function
plt.figure(figsize=(10, 6))
plt.plot(x_values, sigmoid_values, label='Sigmoid Function', color='blue')
plt.title('Sigmoid Function')
plt.xlabel('x')
plt.ylabel('S(x)')
plt.grid(True)
plt.ylim(-0.1, 1.1)  # Extend y-axis to show the asymptotic behavior clearly
plt.axhline(y=0, color='black',linewidth=0.5)
plt.axhline(y=1, color='black',linewidth=0.5)
plt.axvline(x=0, color='black',linewidth=0.5)
plt.show()