Exponential Distribution Calculator (with Graph)
Find waiting-time probabilities and percentiles for the exponential distribution.
What the exponential distribution models
The exponential distribution describes the waiting time until the next event in a process where events happen independently at a constant average rate λ. Examples are the time between customer arrivals, the lifetime of a component that does not wear out, and the time until the next phone call. It is the continuous partner of the Poisson distribution: if events per hour are Poisson with rate λ, the hours between events are exponential with rate λ.
The formulas
- Density: f(x) = λ × e^(−λx), for x ≥ 0.
- P(X ≤ x) = 1 − e^(−λx), and P(X > x) = e^(−λx).
- Mean = 1 ÷ λ, standard deviation = 1 ÷ λ, median = ln 2 ÷ λ.
- The x with a given probability p below it is x = −ln(1 − p) ÷ λ.
Worked examples (mean wait of 10 minutes, so λ = 0.1 per minute)
| Question | Answer |
|---|---|
| Wait of 5 minutes or less | 0.3935 |
| Wait of more than 15 minutes | 0.2231 |
| Wait between 5 and 15 minutes | 0.3834 |
| Median wait | 6.93 minutes |
| 90% of waits are shorter than | 23.03 minutes |
The graph under the result highlights the event you chose, and has Download PNG and Download SVG buttons for reports and slides.
The memoryless property
If you have already waited 10 minutes, the chance of waiting a further 5 is the same as it was at the start. For real systems, the exponential is a good model only when the event rate really is constant. Components that wear out, or queues with rush hours, fit other distributions better.
How to cite this page
CalcPulse. (2026). Exponential Distribution Calculator. https://calcpulse.com/exponential-distribution-calculator/
Frequently asked questions
What is an exponential distribution?
A continuous distribution for the time until the next event, when events occur independently at a constant average rate.
What is the relationship between the exponential and Poisson distributions?
If the number of events per period is Poisson with rate λ, the time between events is exponential with rate λ.
What is the mean of an exponential distribution?
1 ÷ λ. The standard deviation is the same, and the median is ln 2 ÷ λ, about 69% of the mean.
What does memoryless mean?
The chance of waiting a further time t does not depend on how long you have already waited.
How do I find a percentile?
Use x = −ln(1 − p) ÷ λ. The last tab does this.
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