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Exponential Distribution Calculator

Probability that a waiting time falls below, above or between values for an exponential distribution with a given rate λ or mean.

Rate λ (events per unit time)

Time x

Upper time for “between” (optional, > x)

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How it works

F(x) = 1 − e^(−λx) · f(x) = λe^(−λx) · mean = SD = 1/λ

The exponential distribution models the time between events in a Poisson process — calls to a help desk, radioactive decays, component failures. It is memoryless: having already waited t units does not change the distribution of the remaining wait.

Worked example

Calls arrive at λ = 0.5 per minute. P(next call within 3 minutes) = 1 − e^(−1.5) = 0.777; mean gap = 2 minutes.

Frequently asked questions

I know the mean, not the rate.

λ = 1 ÷ mean. A mean wait of 4 minutes means λ = 0.25 per minute.

How does it relate to Poisson?

If event counts per interval are Poisson(λ), the gaps between events are Exponential(λ).

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For education — check assumptions (sample size, distribution) before using statistics in real decisions.