Exponential Distribution Calculator
The Waiting-Time Distribution
If events happen at a steady average rate — the kind the Poisson distribution models — the exponential distribution answers a related but different question: how long until the next one happens? It's the standard model for time between arrivals, component lifetimes, and any process that has no "memory" of how long it's already been waiting.
The Formulas
CDF: P(X ≤ x) = 1 − e^(−λx)
Survival: P(X > x) = e^(−λx)
Mean = 1 ÷ λ
λ (lambda) is the rate parameter — the same average rate used in a Poisson model of the same process. The mean of the distribution is simply its reciprocal.
Worked Example: Equipment Lifetime
A component has a failure rate of λ = 0.5 per year, giving a mean lifetime of 2 years:
| Years (x) | P(fails by x) | P(survives past x) |
|---|---|---|
| 1 | 39.35% | 60.65% |
| 2 | 63.21% | 36.79% |
| 3 | 77.69% | 22.31% |
| 4 | 86.47% | 13.53% |
Notice that even though the mean lifetime is 2 years, only about 63% of units have failed by then — the exponential distribution is right-skewed, so the mean sits above the 50th percentile.
Where This Matters
- Reliability engineering — modeling time-to-failure for components with a constant hazard rate.
- Queueing theory — modeling the time between customer arrivals or service completions.
- Insurance and finance — modeling time between claims or defaults in some risk models.
- Physics and radioactive decay — the memoryless property makes it a natural fit for decay-type processes.
How to Use This Calculator
- Choose Probability Density (PDF), Cumulative Probability (CDF), or Survival Probability from the dropdown.
- Enter the rate (λ).
- Enter the x value you're evaluating.
- Select Calculate to get the result, along with the distribution's mean.
Related Calculations
This distribution describes the gaps between events counted by the Poisson Distribution Calculator. For probabilities over a fixed range with no varying rate, see the Uniform Distribution Calculator.