The distribution of waiting
If events arrive as a Poisson process, the gaps between them follow the exponential distribution. Its one parameter is the rate λ (or equivalently the mean wait 1/λ), and its personality is memorylessness: a component that has survived 100 hours has the same distribution of remaining life as a fresh one. It does not age or wear. The past buys no information.
Mean wait 10 minutes (λ = 0.1). Served within 5 minutes: 1 − e−0.5 = 0.3935. Still waiting after 15: e−1.5 = 0.2231. The median wait is 6.93 minutes, less than the mean, because the long right tail of unlucky waits drags the average up while half of everyone is served faster.
Mean equals SD: a diagnostic
Like its Poisson partner, the exponential carries a fingerprint: mean and standard deviation are equal. Real waiting data with SD far above the mean suggests the rate varies (quiet and rush periods mixed); far below suggests scheduling, not randomness.
Applying memorylessness to things that age. Light bulbs with wearing filaments and humans are not exponential: their remaining-life distribution genuinely changes with survival. The exponential models arrivals and radioactive decay, not wear-out.
Frequently asked questions
How are Poisson and exponential related?
Two views of one process: Poisson counts events per interval; exponential measures the gaps. λ is the same rate in both.