The exponential distribution models the waiting time until the next event in a process where events occur independently at a constant average rate — customer arrivals, equipment failures, radioactive decay, or time between phone calls. This calculator computes the probability of waiting at most, more than, or between specific amounts of time, along with the distribution's mean, median, and variance.
How the Exponential Distribution Calculator works
The calculator uses the exponential cumulative distribution function F(x) = 1 − e^(−λx), where λ (lambda) is the rate parameter — the average number of events per unit of time. P(X ≤ x) is read directly from F(x); P(X > x) is the complement, 1 − F(x); and the range probability P(a < X < b) is F(b) − F(a). The underlying assumption is a Poisson process: events happen independently and at a constant average rate, with no memory of past events.
The mean (μ = 1/λ), median (ln(2)/λ), and variance (1/λ²) are all derived directly from λ. Because the exponential distribution is right-skewed, the median is always less than the mean — most waits are shorter than average, balanced by a smaller number of much longer waits.
Inputs and what they mean
Rate (λ): the average number of events per unit of time, expressed in events per minute, per year, or whatever time unit fits your scenario — it must be greater than zero. x: the specific waiting time you're evaluating a probability for, in the same time unit as λ. Probability type: choose "At most x" for P(X ≤ x), "More than x" for P(X > x), or "A range" to enter a lower bound (a) and upper bound (b) for P(a < X < b). The rate parameter has the largest effect on every output — doubling λ halves the mean, median, and variance.
Limits and edge cases
The exponential distribution assumes a constant event rate over time — it does not fit processes that speed up, slow down, or have a minimum delay (like manufacturing processes with a fixed setup time before failures can occur). If wear-and-tear or aging makes failures more likely over time, a Weibull distribution is usually a better fit. The exponential distribution also only models a single event's waiting time; for the time until the k-th event in a Poisson process, use a Gamma or Erlang distribution instead. Because of the memoryless property, this calculator should not be used for scenarios where past elapsed time changes the odds of the next event — most real-world mechanical wear processes violate that assumption.