--- title: Visualizing Statistical Distributions with Python date: 2024-11-27T08:31:51Z modified: 2026-06-22T16:17:39Z permalink: "https://www.micheledpierri.com/2024/11/27/visualizing-statistical-distributions-with-python/" type: post status: publish excerpt: "" wpid: 927 categories: - Programming tags: - Programming - Data Visualization - Matplotlib - Python featured_image: "https://www.micheledpierri.com/wp-content/uploads/2024/11/draw_curves_.png" featured_image_alt: A barefoot child crouches in a warm, painterly old stone alley, drawing a large bird with white chalk amid cracked ochre walls, wooden chairs, arched doorways, and rising steps. timestamp: 2026-06-22T16:17:39Z --- This post illustrates techniques for visualizing statistical distributions using Python and its graphics libraries, particularly [Matplotlib](https://www.micheledpierri.com/wp-content/uploads/wp-mfa-exports/post/matplotlib-1.md). The resulting charts are used in the [statistical distribution](https://www.micheledpierri.com/elements-of-statistics-for-doctors/statistical-distributions/)s lesson of the [statistics course](https://www.micheledpierri.com/wp-content/uploads/wp-mfa-exports/page/statistics.md). # Required Libraries Import import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm, binom, poisson, expon, uniform, bernoulli, chi2, t ``` import numpy as np import matplotlib.pyplot as plt from scipy.stats import norm, binom, poisson, expon, uniform, bernoulli, chi2, t ``` ## Normal distribution # 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() ``` # 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 # 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() ``` # 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() ``` ## Bernoulli distribution # 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() ``` # 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 # 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() ``` # 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 # 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() ``` # 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 # 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() ``` # 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() ``` ## Chi square distribution # 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 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() ``` ## t distribution vs normal distribution # 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() ``` # 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() ``` ## Sigmoid function # 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() ``` # 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() ```