Exponential Distribution in NumPy

The Exponential Distribution is a continuous probability distribution that describes the time between two events in a Poisson process, where events occur independently and at a constant average rate. NumPy provides a simple method to generate such random values: numpy.random.exponential().

Example: This example shows how to generate one exponential random value using the default parameters.

import numpy as np
x = np.random.exponential()
print(x)

0.5339358426948082

Explanation:

• np.random.exponential() generates one value following the exponential distribution.
• Since no parameters are passed, it uses scale = 1 by default.

Syntax

numpy.random.exponential(scale=1.0, size=None)

Parameters:

• scale: Inverse of the event rate (β = 1/λ).
• size: Shape of output array.

Examples

Example 1: This example generates one exponential random value using a custom scale.

import numpy as np
x = np.random.exponential(scale=2)
print(x)

0.8177243559186411

Explanation:

• scale=2 values will be more spread out.
• x holds a single exponential random number.
• Larger scale values make the distribution longer and wider.

Example 2: This example generates five random numbers from the exponential distribution.

import numpy as np
arr = np.random.exponential(scale=1.5, size=5)
print(arr)

[2.14106221 1.93254045 0.03957526 0.58763751 1.12814399]

Explanation

• scale=1.5 moderate spread.
• size=5 returns 5 values.
• arr stores the array like [0.21, 1.33, 0.94, ...].

Visualizing the Exponential Distribution

Visualizing the generated numbers helps in understanding their behavior. Below is an example of plotting a histogram of random numbers generated using numpy.random.exponential.

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

s = 2      # scale
n = 800    # number of points

data = np.random.exponential(scale=s, size=n)
sns.histplot(data, bins=30, kde=True, edgecolor='black')

plt.title(f"Exponential Distribution (Scale={s})")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.grid(True)
plt.show()

Output

Explanation:

• s = 2 sets the spread of the distribution.
• n = 800 creates enough data points for a smooth histogram.
• sns.histplot() shows: Bars -> simulated data and Curve (kde) -> smooth theoretical shape
• The graph shows high frequency near 0 and a long decreasing tail, which is typical of exponential distributions.