Exponential Distribution Calculator

Background.

Waiting times take on a recognisable shape when the events behind them happen at random at a steady average rate. Calls to a help desk, customers walking into a bank, decays of radioactive atoms, failures of electronic parts that do not wear out: short gaps are common, long gaps are rare, and the chance of a gap stretching past any given length shrinks by the same factor for every extra unit of time. That shape is the exponential distribution, and a single number pins it down completely, the rate λ, meaning the average number of events per unit of time.

Given λ and a time x, three questions have answers. Has the event happened by x? The probability is 1 − e^(−λx). Are we still waiting at x? That is the complement, e^(−λx). And how densely is probability packed around x? That is λe^(−λx), a rate per unit of time rather than a probability.

A few facts make the distribution easy to reason with. The mean wait is 1/λ, so a rate of 0.2 per minute means a call every 5 minutes on average. The chance that a wait is over by the mean is always 1 − 1/e, about 63.2%, whatever λ is, so most waits are shorter than average and the long tail drags the mean up. The median sits at ln 2 / λ, about 69% of the mean. And the process has no memory: after 5 minutes without a call, the chance of waiting 5 more is exactly what it was at the start.

That last property is also the model's limit. Anything that ages, wears out, runs to a timetable or bunches up after a quiet spell breaks it. The page also takes λ as given; it does not estimate it from data, test whether your data are exponential, or handle observations cut off before the event occurred.

What is exponential distribution calculator?

The exponential distribution is the continuous probability distribution of the waiting time between events in a Poisson process, where events occur independently at a constant average rate λ. Its density is f(x) = λe^(−λx) for x ≥ 0, its cumulative distribution is F(x) = 1 − e^(−λx), and its survival function is S(x) = e^(−λx). The mean and the standard deviation are both 1/λ. It is the only continuous distribution without memory, which is why it underlies constant-failure-rate reliability models and radioactive decay.

How to use this calculator.

1. 01Enter the rate λ as events per unit of time, such as 0.2 calls per minute or 0.001 failures per hour. If you know the mean waiting time instead, enter its reciprocal: a mean of 5 minutes is λ = 0.2.
2. 02Enter the time x in the same unit the rate uses. A rate per hour with a time in minutes gives an answer that looks reasonable and is wrong by a factor of 60.
3. 03Read P(X ≤ x) as the chance the event has occurred by time x, and P(X > x) as the chance you are still waiting. The two always add to 1.
4. 04For the chance that the event falls between two times a and b, run the page at both and subtract the cumulative values: F(b) − F(a).

The formula.

With a rate λ > 0 and a time x ≥ 0, F(x) = 1 − e^(−λx), S(x) = e^(−λx) and f(x) = λe^(−λx). Everything depends on the product λx, the expected number of events in time x. When λx = 1, x equals the mean and F = 1 − e^(−1) = 0.632; when λx = ln 2 ≈ 0.693, F = 0.5 and x is the median; by λx = 3, F has reached 0.950. The density is the slope of F: it starts at λ when x = 0 and falls by a factor of e for every 1/λ of elapsed time. It is measured in reciprocal time units, which is why it can exceed 1 when λ is large; only areas under it are probabilities. The rate must be strictly positive. A time of zero is allowed and gives F = 0 and S = 1.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST/SEMATECH e-Handbook, Exponential Distribution.
2. [2]OpenStax, Introductory Statistics 2e, section 5.3, The Exponential Distribution.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.