Table of Contents
The Exponential Distribution
The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is the continuous analog of the geometric distribution and the only continuous distribution with the memoryless property.
Common applications include modeling customer arrival times, component lifetimes, radioactive decay intervals, and time between earthquakes. Its simplicity and mathematical elegance make it a cornerstone of reliability engineering and queueing theory.
Formulas
Properties
| Property | Value |
|---|---|
| Mean | 1/λ |
| Median | ln(2)/λ |
| Mode | 0 |
| Variance | 1/λ² |
| Skewness | 2 |
Examples
If a server processes requests at rate λ=0.5 per minute, the mean wait is 1/0.5 = 2 minutes. P(wait ≤ 3 min) = 1 - e^(-0.5×3) = 77.7%.
Frequently Asked Questions
What is the memoryless property?
P(X > s+t | X > s) = P(X > t). If you have already waited s minutes, the probability of waiting at least t more minutes is the same as waiting t minutes from the start. The past does not affect the future.
When should I not use exponential?
Do not use it when the hazard rate changes over time (e.g., aging components with increasing failure rates). Use Weibull or log-normal distributions instead for non-constant hazard rates.
Relationship to Poisson?
If events follow a Poisson process with rate λ, the time between consecutive events follows Exp(λ). They are complementary views of the same process.