Exponential Distribution Calculator
Find probabilities for waiting times between events.
What the Exponential Distribution Calculator does
The exponential distribution models the time until the next event when events occur at a constant average rate. It is the continuous counterpart to the Poisson distribution and the standard model for component lifetimes and queue waiting times.
Formula
P(X ≤ x) = 1 − e^(−λx)Probability density = λ × e^(−λx)Mean = Standard deviation = 1 ÷ λ
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Specify by | one of 2 options | Yes | — |
| Rate (λ) | number | In some modes | Accepts more than 0. Shown Specify by is Rate (λ) — events per unit time. |
| Mean waiting time | number | In some modes | Accepts more than 0. Shown Specify by is Mean waiting time. |
| Time value (x) | number | Yes | Accepts 0 or more. |
How to use it
- Choose Specify by.
- Enter Time value (x).
- Fill in the remaining inputs the form shows for your choice.
- Select Calculate.
Worked example
Buses arrive on average every 12 minutes — what is the chance of waiting 5 minutes or less?
- Mode
- Mean
- Mean time
- 12
- x
- 5
λ = 1/12, so P = 1 − e^(−5/12) = 0.3408, about 34.1%.
Frequently asked questions
What does memoryless mean?
The probability of waiting another 5 minutes is the same whether you just arrived or have already waited an hour. The distribution has no memory of elapsed time.
Is it realistic for real waiting times?
It fits well when arrivals are genuinely random. Scheduled services like trains are not exponential, because arrivals cluster around the timetable.