Exponential Distribution Calculator
The distribution of waiting times between random events — probabilities, mean, median and the memoryless property that makes it unique.
If events arrive at a constant average rate λ, the gap between consecutive events is exponentially distributed: P(X > t) = e^(−λt).
How the exponential distribution calculator works
If events arrive at a constant average rate λ, the gap between consecutive events is exponentially distributed: P(X > t) = e^(−λt). The mean wait is 1/λ.
Its defining feature is memorylessness: having waited ten minutes already tells you nothing about how much longer you will wait. It is the only continuous distribution with that property, and it is exactly why the exponential is the wrong model for anything that wears out.
Formula: P(X ≤ t) = 1 − e^(−λt); mean = 1/λ
Worked examples
| Inputs | P(wait ≤ t) | Note |
|---|---|---|
| Mean 10, wait 5 | 39.346934% | 39.3% chance of a wait under 5 |
| Waiting one mean | 63.212056% | 63.2% — always, whatever the rate |
| The median | 49.997641% | 50% — the median is ln2 ÷ λ |
FAQFrequently asked questions
What does the exponential distribution model?
The time between independent events happening at a constant average rate — calls to a helpdesk, radioactive decays, arrivals at a queue.
What is the memoryless property?
The chance of waiting another 10 minutes is the same whether you have waited 0 minutes or 100. The exponential is the only continuous distribution with this property.
Does that make it wrong for equipment failure?
For anything that wears out, yes. Memorylessness says a 20-year-old machine is as good as new. The Weibull distribution handles ageing properly.
Why is the median below the mean?
Because the distribution is strongly right-skewed. The median is ln2 ÷ λ, about 69% of the mean.
How does it relate to the Poisson distribution?
Two views of the same process: Poisson counts events in a fixed interval, exponential measures the gaps between them.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.