Exponential distribution
Model waiting time between constant-rate Poisson events.
Exponential distribution is one of 7 distributions formulas in the probability section of this library, and it is used at university level.
Why exponential distribution works
It describes the waiting time until the next event when events arrive at a steady rate. The survival expression is exactly the Poisson probability of seeing zero events in that stretch of time, which is why waiting times and event counts are two views of the same underlying process.
What each symbol means
$\lambda$ is the positive event rate.
Exponential distribution: when it holds
$x\ge0$, $\lambda>0$, and the model has the memoryless property.
When it stops applying
It assumes memorylessness, meaning age never matters. That is wrong for anything that wears out, such as a machine part or a living organism, where the chance of failing in the next hour rises with time already survived. Those situations need a distribution such as the Weibull instead.
Exponential distribution: a worked example
For $\lambda=2$, $P(X>1)=e^{-2}$.
The mistake to avoid
What people do: Swapping the rate with the mean waiting time.
Why it goes wrong: A rate of 2 events per hour means an average wait of half an hour, not two hours. Substituting the mean where the rate belongs inverts every answer the model produces.
Do this instead: Check the units: the rate is events per unit of time and the mean is time per event, so one is always the reciprocal of the other.
Exponential distribution: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\lambda$ is the positive event rate.
- Check the conditions before substituting. $x\ge0$, $\lambda>0$, and the model has the memoryless property.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where exponential distribution comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about exponential distribution
What does memoryless mean in practice?
That having already waited ten minutes tells you nothing about how much longer you will wait. The remaining time has exactly the same distribution it had when you started waiting.
How is this connected to the Poisson distribution?
They describe the same process from two angles. Poisson counts how many events land in a fixed interval, while this measures the gaps between consecutive events.
Is the median the same as the mean?
No, the distribution is skewed so the median is smaller. With a rate of 2, the mean wait is 0.5 while the median is the natural log of 2 divided by 2, about 0.347.
Can the variable take a negative value?
No, waiting time starts at zero. The density is defined only for values of zero or more, which is one way this differs from the normal distribution.