Poisson Distribution in NumPy

The Poisson Distribution models how many times an event occurs within a fixed interval when the average occurrence rate (λ) is known. It is commonly used for scenarios like customer arrivals, call center traffic, website visits, or any event that happens independently over time or space.

In NumPy, we use the numpy.random.poisson() method to generate Poisson-distributed random values.

Example: In this example, we generate a basic Poisson-distributed number using the default parameters to understand how the function works.

import numpy as np
x = np.random.poisson(lam=3)
print(x)

5

Explanation: np.random.poisson(lam=3) generates numbers around λ = 3, which is the expected average event count.

Syntax

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

Parameters:

• lam: Average expected events (λ).
• size: Shape of output (e.g., single value, 1D array, 2D array).

Examples

Example 1: In this example, we generate one random number with an expected average of λ = 5.

import numpy as np
x = np.random.poisson(lam=5)
print(x)

5

Explanation: np.random.poisson(lam=5) returns a number whose expected average value is 5.

Example 2: Here, we generate five Poisson-distributed numbers using λ = 5.

import numpy as np
arr = np.random.poisson(lam=5, size=5)
print(arr)

[4 9 4 5 3]

Explanation: size=5 creates a 1D array of 5 Poisson values generated using λ = 5.

Example 3: In this example, we generate a 2×3 array of Poisson values using λ = 4.

import numpy as np
m = np.random.poisson(lam=4, size=(2, 3))
print(m)

[[6 6 3]
 [6 2 8]]

Explanation: size=(2, 3) tells NumPy to generate a 2×3 matrix filled with Poisson-distributed values.

Visualizing the Poisson Distribution

To understand the distribution better we can visualize the generated numbers. Here is an example of plotting a histogram of random numbers generated using numpy.random.poisson.

import numpy as np
import matplotlib.pyplot as plt

lam = 3
data = np.random.poisson(lam=lam, size=1000)

plt.hist(data, bins=np.arange(-0.5, max(data)+1.5, 1), edgecolor='black', density=True)
plt.title(f"Poisson Distribution (λ={lam})")
plt.xlabel("Number of Events")
plt.ylabel("Probability")
plt.grid(True)
plt.show()

Output

Explanation:

• np.random.poisson(lam, size=1000) generates 1000 values following the Poisson distribution.
• The histogram shows how frequently each event count appears (0, 1, 2, …).
• Poisson histograms usually peak near λ and decrease as the event count increases.