Exponential distribution calculator

What is the exponential distribution?

The distribution of the time between events that happen independently at a constant average rate λ — waiting times, time between calls, the life of a part that does not wear out. P(X < x) = 1 − e^(−λx), P(X > x) = e^(−λx), and the density is f(x) = λe^(−λx).

Example: a mean of 10 years

With μ = 10 the rate is λ = 0.1. The chance of lasting more than 7 years is e^(−0.7) = 0.4966; the 80th percentile is −ln(1 − 0.8) ÷ 0.1 = 16.1 years; and the chance of failing between 9 and 11 years is 0.0737 (OpenStax Introductory Statistics 2e, Example 5.9).

Rate or mean — which do I enter?

Either: λ = 1 ÷ μ. A mean wait of 4 minutes is a rate of 0.25 per minute. The standard deviation equals the mean, and the median, ln 2 ÷ λ ≈ 0.693 × μ, is below the mean — most waits are short, a few are long.

What does “memoryless” mean?

Time already spent waiting does not change what is left: P(X > s + t | X > s) = P(X > t). If waits average 4 minutes and you have already waited 4, the chance of waiting more than 3 more minutes is still e^(−0.75) = 0.4724 (OpenStax Example 5.12).

How is it related to the Poisson distribution?

They describe the same process from two sides. If the number of events per unit of time is Poisson with mean λ, the time between events is exponential with mean 1 ÷ λ. Four calls a minute on average means an average of 0.25 minutes between calls.

Where do the formulas come from?

From OpenStax Introductory Statistics 2e, section 5.3 “The Exponential Distribution”: the density, cumulative probabilities, percentiles and the memoryless property. Its worked examples are part of this calculator’s automated tests.

Is the exponential distribution a good model for how long a machine lasts?

Only while the failure rate really is constant. The exponential has a constant hazard, so it fits electronics in their stable middle period or events like random arrivals, but not parts that wear out, where the chance of failure rises with age. If the failures cluster at old age, a distribution such as the Weibull is the usual choice.

Why is the median smaller than the mean?

The distribution is skewed to the right: many short waits and a few very long ones pull the mean up. The median is ln 2 ÷ λ, about 69% of the mean. For a mean wait of 6 minutes the median is 4.16 minutes, so half of the waits are shorter than that and about 36.8% last longer than the mean.

Formulas and units

With rate λ (events per unit of time) the mean and standard deviation are both 1 ÷ λ, the variance is 1 ÷ λ², the probability of waiting no longer than x is 1 − e^(−λx), and the probability of waiting longer than x is e^(−λx). The quantile for probability q is −ln(1 − q) ÷ λ. The unit of λ is the reciprocal of the unit of x: if x is in minutes, λ is per minute. Mixing the two is the most common source of wrong answers.

Worked example: a mean wait of 6 minutes

Suppose the wait for a bus averages 6 minutes, so λ = 1 ÷ 6 = 0.1667 per minute. The chance of waiting more than 10 minutes is e^(−10 ÷ 6) = 0.1889. The chance the wait falls between 5 and 8 minutes is 0.1710. The 90th percentile is −ln(0.1) × 6 = 13.82 minutes. By the memoryless property, if you have already waited 6 minutes, the chance of waiting at least 4 more is still e^(−4 ÷ 6) = 0.5134, the same as for a fresh arrival.

Common mistakes

Entering the mean where the rate is expected (or the reverse) is the first: a mean of 4 minutes is a rate of 0.25 per minute, not 4. The second is applying the distribution to a process whose events are not independent or whose rate changes over time, such as a bus that runs on a timetable or traffic that peaks at rush hour. Third, a sample mean is only an estimate of μ; with few observations the rate is uncertain, and the computed probabilities inherit that uncertainty.

Sources: OpenStax Introductory Statistics 2e, section 5.3 (the exponential distribution, its mean, percentiles and the memoryless property). Related: Poisson distribution calculator, normal distribution calculator, geometric distribution calculator and half-life calculator.