exponential distribution
The exponential distribution models the waiting time until the next random event, when those events arrive at a steady average rate. How long until the next claim lands? How long until a machine breaks down? If events happen Poisson-style — at a constant rate, independently — then the gaps between them follow an exponential distribution. It is the continuous twin of the geometric distribution.
It has a single rate parameter λ (lambda); equivalently a mean waiting time of 1/λ. Its density falls away exponentially: short waits are most likely and long waits become rapidly rarer, but never impossible. Its defining property is memorylessness — the only continuous distribution with it. A component that has already run for 10 hours has exactly the same future failure prospects as a brand-new one; the past leaves no trace. Mathematically, P(wait more than s+t given already waited s) = P(wait more than t).
Actuaries use the exponential as a first model for time-to-claim and time-to-failure, and as the simplest severity distribution for claim sizes. Its memorylessness corresponds to a constant force of mortality or a constant hazard rate — a tidy assumption that makes the math clean. But that same property is usually its downfall in practice: real lifetimes, machines, and policies almost always age, so their failure rate rises over time, which the exponential cannot represent. That is precisely why the Weibull and gamma, which allow a changing hazard, exist.
If claims arrive at rate λ = 2 per week, the average gap between claims is 1/2 = half a week. The chance the next claim takes more than a full week to arrive is e^(-2) ≈ 0.135.
The exponential turns a Poisson event rate into the waiting time between events.
Memorylessness means the exponential cannot model wear-out or aging — using it for human lifetimes or aging equipment understates risk at older ages, which is why a constant-hazard assumption is rarely realistic over a long span.