Poisson Distribution

📊 Poisson Distribution in Python

The Poisson Distribution models the number of events occurring in a fixed interval of time or space, assuming the events happen independently at a constant rate.

It’s widely used in traffic flow, call center arrivals, and natural event counts.

1. Characteristics of Poisson Distribution

λ (lam) → expected number of events per interval
size → number of random samples
Values are non-negative integers: 0,1,2,…
Probability Mass Function (PMF):

$\frac{e^{-\lambda} \lambda^k}{k!}$

Where $k = 0, 1, 2, \dots$

2. Generate Poisson Data Using NumPy

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns# Parameters
lam = 5 # expected number of events
size = 1000 # number of samples

# Generate random Poisson data
data = np.random.poisson(lam, size)

print(data[:10]) # first 10 values

import numpy as np

import matplotlib.pyplot as plt

import seaborn as sns# Parameters

lam = 5 # expected number of events

size = 1000 # number of samples

# Generate random Poisson data

data = np.random.poisson(lam, size)

print(data[:10]) # first 10 values

Output (example):

3. Visualize Poisson Distribution

Peaks near λ
Discrete integer values

4. Change λ to See Effect

data_small = np.random.poisson(lam=2, size=1000)
data_large = np.random.poisson(lam=10, size=1000)sns.histplot(data_small, bins=range(0, max(data_small)+2), color='red', label='λ=2', kde=False)
sns.histplot(data_large, bins=range(0, max(data_large)+2), color='blue', label='λ=10', kde=False)
plt.legend()
plt.title('Poisson Distribution Comparison')
plt.show()

data_small = np.random.poisson(lam=2, size=1000)

data_large = np.random.poisson(lam=10, size=1000)sns.histplot(data_small, bins=range(0, max(data_small)+2), color='red', label='λ=2', kde=False)

sns.histplot(data_large, bins=range(0, max(data_large)+2), color='blue', label='λ=10', kde=False)

plt.legend()

plt.title('Poisson Distribution Comparison')

plt.show()

Smaller λ → peak closer to 0
Larger λ → distribution spreads and peak moves right

5. Compute Mean and Variance

Theoretical property of Poisson:

Mean = Variance = λ

🎯 Practice Exercises

Model number of calls per hour with λ=3 for 1000 hours.
Generate Poisson data with λ=7 and plot histogram.
Compare λ=2, λ=5, λ=10 in one plot.

Poisson Distribution