Exponential Distribution

The Exponential Distribution is one of the most commonly used probability distributions in statistics and data science. It is widely used to model the time or space between events in a Poisson process. In simple terms, it describes how long you have to wait before something happens, like a bus arriving or a customer calling a help center.

For example, if buses arrive at a bus stop every 15 minutes on average, the time you wait for the next bus can be modeled using an exponential distribution.

Probability Density Function (PDF)

The probability density function of the exponential distribution is:

f(x; \lambda) = \lambda e^{-\lambda x}, \quad x \geq 0

Where:

• \lambda > 0 is the rate parameter (how often events occur)
• x is the time or distance until the next event

Cumulative Distribution Function (CDF)

The cumulative distribution function gives the probability that the event occurs within time :

F(x; \lambda) = 1 - e^{-\lambda x}, \quad x \geq 0

Mean and Variance

• Mean (Expected Value): E[X] = \frac{1}{\lambda}
• Variance: \text{Var}(X) = \frac{1}{\lambda^2}

Memoryless Property

The exponential distribution is memoryless, which means:

P(X > s + t \mid X > s) = P(X > t)

This property tells us that the probability of waiting longer does not depend on how long you've already waited. This is unique to the exponential distribution.

Example

Suppose calls come into a customer support center at an average rate of 2 per minute. What is the probability that you wait more than 30 seconds for the next call?

Solution:

1. Understand the Rate (λ):

• Since 2 calls come in per minute, that means the average rate is: λ=2 calls per minute

2. Convert Time:

• Find the probability of waiting more than 30 seconds. But since the rate is in minutes, convert 30 seconds to minutes: 30 seconds=0.5 minutes

3. Use the Exponential Distribution formula

• P(X > 0.5) = e^{-\lambda x} = e^{-2 \cdot 0.5} = e^{-1} \approx 0.3679

So, there is about a 36.79% chance that the next call comes after 30 seconds.

Example in Python

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

# Set the rate parameter lambda
lam = 1.5

# Generate exponential data
data = np.random.exponential(1/lam, 1000)

# Plot the KDE and histogram
sns.histplot(data, kde=True, stat="density")
plt.title("Exponential Distribution (lambda=1.5)")
plt.xlabel("Time")
plt.ylabel("Density")
plt.show()

Output:

Relation to the Poisson Process

The exponential distribution is closely related to the Poisson distribution. If events occur following a Poisson process with rate , then the time between events is exponentially distributed with parameter .

Real-World Applications

1. Call Centers: Time between incoming calls
2. Bank Queues: Time between customer arrivals
3. Web Servers: Time between requests
4. Manufacturing: Time until a machine part fails
5. Transport: Time between buses or trains