Exponential distribution calculator
Fits waiting times between random events, such as the time until the next customer arrives or the next call reaches a call centre.
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How to use it
- Enter i know the, value, time x and second time x2 (range only).
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
P(X ≤ x) = 1 - e^(-λx), P(X > x) = e^(-λx), P(x1 < X ≤ x2) = e^(-λ × x1) - e^(-λ × x2). Mean = 1 ÷ λ, median = ln 2 ÷ λ, standard deviation = 1 ÷ λ.
Example
Inputs: I know the = Rate λ (events per time unit), Value = 0.5, Time x = 2, Second time x2 (range only) = 4. Result: P(X ≤ x), event within x: 0.6321205588; P(X > x), wait longer than x: 0.3678794412; Rate λ: 0.5.
Limits and caveats
- Assumes events happen at random at a constant rate with no memory, which does not fit wear-out failures that rise with age.
- Use the same time unit for the rate and x: a rate per hour needs x in hours.
Questions about this tool
How does it relate to the Poisson distribution?
If the count of events in an interval is Poisson with rate λ, the time between consecutive events is exponential with the same rate.
Why is the median below the mean?
The distribution is skewed right: most waits are short and a few very long ones pull the mean up. The median is 0.693 of the mean.
What does memoryless mean?
If you have waited 5 minutes with no arrival, the chance of waiting 2 more minutes is the same as it was at the start.