The Poisson Distribution is a fundamental statistical tool used to model the number of times an event occurs within a fixed interval of time or space, given a known average rate and independence between events.

In Python, we can easily simulate and analyze Poisson-distributed data using NumPy.

What is the Poisson Distribution?

The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Probability Mass Function (PMF)

P(X=k)=e−λ⋅λkk!P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}

Where:

• kk: Number of events (0, 1, 2, ...)

• λ\lambda: Average number of events in a fixed interval

• ee: Euler's number (~2.71828)

Real-Life Examples

NumPy’s Poisson Function

Syntax

numpy.random.Generator.poisson(lam=1.0, size=None)

Parameters

✅ Returns

Random samples following a Poisson distribution.

✅ Example: Basic Usage

import numpy as np

# Create a random generator
rng = np.random.default_rng(seed=42)

# Simulate 1000 events with λ = 5
data = rng.poisson(lam=5, size=1000)

Visualizing Poisson Distribution

import seaborn as sns
import matplotlib.pyplot as plt

sns.histplot(data, bins=range(0, max(data)+2), discrete=True, color='skyblue')
plt.title("Poisson Distribution (λ = 5)")
plt.xlabel("Number of Events")
plt.ylabel("Frequency")
plt.show()

This histogram shows how often each count of events occurred across the 1000 simulations.

Varying λ Values

λ = 2 (Low Rate)

data = rng.poisson(lam=2, size=1000)
sns.histplot(data, bins=range(0, 10), discrete=True, color='lightgreen')
plt.title("Poisson Distribution (λ = 2)")
plt.show()

λ = 10 (Higher Rate)

data = rng.poisson(lam=10, size=1000)
sns.histplot(data, bins=range(0, 20), discrete=True, color='orange')
plt.title("Poisson Distribution (λ = 10)")
plt.show()

As λ increases, the distribution becomes more symmetric and starts to resemble a normal distribution.

Statistical Properties

For a Poisson distribution:

Mean=λVariance=λ\text{Mean} = \lambda \quad\quad \text{Variance} = \lambda

data = rng.poisson(lam=7, size=10000)
print("Sample Mean:", np.mean(data))
print("Sample Variance:", np.var(data))

Real-World Use Case: Web Server Traffic

# Average 20 requests per second
requests = rng.poisson(lam=20, size=60)  # Simulate 1 minute of traffic

plt.plot(requests, marker='o')
plt.title("Web Server Requests per Second")
plt.xlabel("Second")
plt.ylabel("Requests")
plt.grid(True)
plt.show()

This simulates a traffic pattern that might help with load testing or capacity planning.

Full Code Example

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

# Random generator
rng = np.random.default_rng(seed=123)

# Parameters
lambda_val = 4
sample_size = 1000

# Generate Poisson-distributed data
data = rng.poisson(lam=lambda_val, size=sample_size)

# Plot
sns.histplot(data, bins=range(0, max(data)+2), discrete=True, color='lightblue')
plt.title(f"Poisson Distribution (λ = {lambda_val})")
plt.xlabel("Number of Events")
plt.ylabel("Frequency")
plt.axvline(np.mean(data), color='red', linestyle='--', label='Mean')
plt.legend()
plt.show()

# Stats
print("Sample Mean:", round(np.mean(data), 2))
print("Sample Variance:", round(np.var(data), 2))
print("Expected (λ):", lambda_val)

✅ Tips

1. ✅ Use default_rng() for modern random number generation.

2. ✅ Use size=(n, m) to generate multidimensional data.

3. ✅ Large λ → distribution approaches normal.

4. ✅ Best for rare event modeling over time or space.

5. ✅ Set seed during development for reproducibility.

⚠️ Common Pitfalls

Poisson vs. Binomial vs. Normal

Conclusion

The Poisson distribution is a powerful tool for modeling event counts in a fixed time or space interval. Whether you're monitoring website traffic, predicting system failures, or analyzing arrival rates, NumPy’s poisson() makes simulation easy.