The exponential distribution is a probability distribution used to model the time between events that happen randomly and independently at a constant rate. It’s implemented in Python’s NumPy library using the np.random.exponential function.

Here’s a breakdown of the key points:

Properties:

• Memoryless: The time since the last event doesn’t affect the probability of the next event happening. So, the probability of an event occurring in the next second is always the same, regardless of how much time has passed since the previous event.
• Probability Density Function (PDF): The PDF describes the probability of an event happening at a specific time. In the case of the exponential distribution, the PDF is:

• f(x) = λ * exp(-λx) for x ≥ 0
• Here, λ (lambda) is the rate parameter. It controls how often events occur:

• Higher λ: More frequent events, shorter intervals between events.
• Lower λ: Less frequent events, longer intervals between events.

Applications:

• Simulating waiting times: Customer arrivals at a store, service times, or time between network packets arriving.
• Modeling lifespans: Components or systems that are more likely to fail earlier in their life (e.g., light bulbs).

Generating Random Samples with np.random.exponential:

import numpy as np

# Generate 5 random samples with scale parameter 2.0 (lambda=0.5)
samples = np.random.exponential(scale=2.0, size=5)
print(samples)

Output

[0.00696876 0.89895024 0.29246412 2.65108078 1.34133834]

Parameters:

• scale (float or array-like of floats): The scale parameter (default: 1.0) is related to the rate parameter (λ) by scale = 1 / λ. So, a smaller scale value corresponds to a higher rate (more frequent events).
• size (int or tuple of ints, optional): The desired output shape. If left as None (default), it returns a single value if scale is a scalar, otherwise an array with the same shape as scale.