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¶

```python 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:

```python 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.

```python 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:

```python 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

Exercises¶

Exercise 1. Write code that generates 2000 samples from a normal distribution, plots a histogram with density=True, and overlays the theoretical PDF curve using scipy.stats.norm.pdf().

Exercise 2. Explain why density=True is necessary when overlaying a PDF curve on a histogram. What units does the y-axis represent?

Exercise 3. Create a figure that fits and overlays an exponential distribution PDF on histogram data generated from np.random.exponential().

Exercise 4. Write code that generates data from a mixture of two normal distributions and overlays both individual component PDFs and the combined PDF on the histogram.