Hong Life CalculatorsUnderstanding the exponential distribution calculator — concept and principle
An exponential distribution calculator gives the probability that a waiting time is shorter or longer than a chosen time x, when events happen at a constant average rate λ. It also returns the mean and median waiting time. Typical uses are the time until the next customer arrives, the time between calls to a help desk, the lifetime of components with a constant failure rate, and radioactive decay.
How it is calculated
- P(X ≤ x) = 1 − e^(−λx), the chance the event has happened by time x.
- P(X > x) = e^(−λx), the chance you are still waiting at time x (the survival function).
- Mean = 1/λ, the average waiting time.
- Median = ln 2 ÷ λ ≈ 0.6931 ÷ λ, the time by which half of the waits are over.
λ is the number of events per unit of time, and x must be in that same unit: if λ is per hour, x is in hours. The variance is 1/λ², so the standard deviation equals the mean.
The principle
If events occur independently at a constant rate (a Poisson process), the number of events in time x is Poisson with mean λx. Waiting longer than x means seeing zero events in that time, whose Poisson probability is e^(−λx). That gives the survival function directly. The exponential is the only continuous distribution that is memoryless: P(X > s + t | X > s) = P(X > t). Having already waited does not make the event any more or less likely to happen soon. The median is below the mean because the distribution has a long right tail.
Worked example
- Inputs: λ = 0.5 per unit time, x = 3.
- λx = 0.5 × 3 = 1.5, and e^(−1.5) ≈ 0.22313.
- P(X ≤ 3) = 1 − 0.22313 ≈ 0.7769; P(X > 3) ≈ 0.2231.
- Mean = 1 ÷ 0.5 = 2; median = 0.693147 ÷ 0.5 ≈ 1.3863.
| Rate λ (per unit time) | P(X ≤ x) |
|---|---|
| 0.125 | 0.3127107212 |
| 0.25 | 0.5276334473 |
| 0.375 | 0.6753475326 |
| 0.5 | 0.7768698399 |
| 0.625 | 0.8466450332 |
| 0.75 | 0.8946007754 |
| 1 | 0.9502129316 |
How to read the result
With an average wait of 2 units, there is about a 78% chance the event happens within 3 units. At x equal to the mean, P(X ≤ x) is always 1 − e^(−1) ≈ 63.2%, so most waits are shorter than average. The probability of waiting three times the mean is e^(−3) ≈ 5%.
Common mistakes
- Entering the mean waiting time as λ; λ is its reciprocal.
- Mixing units, such as λ per hour with x in minutes.
- Using the model for wear-out failures, where the failure rate rises with age and a Weibull model fits better.
Frequently asked questions
What does memoryless mean in practice?
If a bus arrives as an exponential process and you have waited 10 minutes, your expected remaining wait is the same as when you arrived.
How do I get λ from data?
Divide the number of events by the total observation time, or take 1 ÷ the average waiting time.
What is the probability between two times a and b?
e^(−λa) − e^(−λb).
Sources: Ross, S. M., A First Course in Probability (Pearson), chapter on continuous random variables · NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.6.6.7 Exponential Distribution