Wait-time between memoryless events
The exponential distribution models the time until the next event in a Poisson process, where events occur randomly at a constant average rate. The distribution is controlled by a rate parameter lambda (or its inverse, the mean wait time).
A key property is memorylessness: the time until the next event is independent of how long it has been since the last event. Regardless of when you start observing, the expected time to the next event is always the same. This property makes the exponential distribution ideal for modeling radioactive decay, server request arrivals, and component lifetimes when wear-out effects are negligible.
Rapid decay and tail behavior
The exponential distribution always has the same characteristic shape: a peak at zero that rapidly decays into a long right tail. Increasing lambda makes the decay steeper and concentrates probability near zero; decreasing lambda flattens the shape and extends the tail.
The mean and standard deviation of an exponential distribution are equal (both 1 divided by lambda), a feature that distinguishes it from other continuous distributions. This equality means the standard deviation is often quite large relative to the mean, reflecting the potential for occasional very long wait times.