Distribution Fitting

A common use case for histograms is visualizing empirical data alongside theoretical probability distributions. This involves estimating distribution parameters from data and overlaying the fitted PDF.

Fitting Normal Distribution

Manual PDF Formula

import matplotlib.pyplot as plt
import numpy as np

def main():
    # data generation
    n_samples = 10_000
    data = np.random.randn(n_samples)  # (10_000,)

    # plot histogram with theoretical PDF (standard normal)
    fig, ax = plt.subplots()
    _, bins, _ = ax.hist(data, bins=100, density=True)
    ax.plot(bins, np.exp(-bins**2 / 2) / np.sqrt(2 * np.pi), 
            '--r', alpha=0.9, lw=5)
    plt.show()

if __name__ == "__main__":
    main()

With Parameter Estimation

When data may not be standard normal, estimate parameters from the sample:

import matplotlib.pyplot as plt
import numpy as np

def main():
    # data generation
    n_samples = 10_000
    data = np.random.randn(n_samples)  # (10_000,)

    # parameter estimation
    mu = data.mean()
    sigma = data.std()

    # plot histogram with fitted PDF
    fig, ax = plt.subplots()
    _, bins, _ = ax.hist(data, bins=100, density=True)
    pdf = np.exp(-(bins - mu)**2 / (2 * sigma**2)) / np.sqrt(2 * np.pi * sigma**2)
    ax.plot(bins, pdf, '--r', alpha=0.9, lw=5)
    plt.show()

if __name__ == "__main__":
    main()

Using scipy.stats

The scipy.stats module provides a cleaner interface for distribution fitting.

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats

def main():
    # data generation from non-standard normal
    n_samples = 10_000
    data = stats.norm(loc=1, scale=2).rvs(n_samples)  # mean=1, std=2

    # parameter estimation
    mu = data.mean()
    sigma = data.std()

    # plot histogram with fitted PDF
    fig, ax = plt.subplots()
    _, bins, _ = ax.hist(data, bins=100, density=True)
    ax.plot(bins, stats.norm(loc=mu, scale=sigma).pdf(bins), 
            '--r', alpha=0.9, lw=5)
    plt.show()

if __name__ == "__main__":
    main()

General Workflow

1. Generate or load data: Obtain the empirical dataset
2. Estimate parameters: Use sample statistics (mean, std) or MLE
3. Plot histogram: Use density=True to normalize
4. Overlay PDF: Evaluate theoretical PDF at bin edges
5. Assess fit: Visual comparison of histogram and fitted curve

Fitting Other Distributions

The same pattern applies to other distributions:

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats

# Exponential distribution
data = stats.expon(scale=2).rvs(10_000)
scale_hat = data.mean()

fig, ax = plt.subplots()
_, bins, _ = ax.hist(data, bins=100, density=True, alpha=0.5)
ax.plot(bins, stats.expon(scale=scale_hat).pdf(bins), '--r', lw=2)
plt.show()

Documentation

• matplotlib.axes.Axes.hist
• scipy.stats