the exponential distribution

How long until the next phone call comes in to a help desk? Until the next radioactive atom decays? Until the next customer walks through the door? When events happen at a steady average rate and don't coordinate with each other, the waiting time until the next one follows the exponential distribution — the continuous cousin of the geometric distribution.

Write X ~ Exponential(lambda), where lambda > 0 is the rate (events per unit time). The density is f(x) = lambda e^(-lambda x) for x >= 0: it starts high at x = 0 and decays smoothly, so short waits are the most likely but very long waits are still possible. The survival function — the chance you wait longer than t — is P(X > t) = e^(-lambda t). The mean wait is E[X] = 1/lambda (faster rate, shorter wait) and the variance is Var(X) = 1/lambda^2. It has a constant hazard rate equal to lambda: at every instant, given you have waited this long, the chance the event happens in the next tiny moment is the same.

The exponential is the bridge to the Poisson process: if events arrive as a Poisson process with rate lambda, the gaps between consecutive events are independent Exponential(lambda) waits. It is the go-to model for lifetimes of things that fail 'randomly' rather than by wearing out, for inter-arrival times in queues, and as the building block of the gamma distribution. One caveat: its memorylessness (next entry) makes it model things that do NOT age — a real lightbulb that wears out is better described by a Weibull distribution.