Calculate probabilities, PDF, CDF, and statistics for exponential distributions.
Last updated: March 2026
Higher λ = shorter expected time between events
The exponential distribution models the time between independent random events occurring at a constant rate. For example, it can represent the waiting time until a new customer arrives, time until a radioactive particle decays, or time to failure of an electronic component.
The distribution is characterized by its rate parameter λ (lambda), which is the average number of events per unit time. A small λ means events occur infrequently (long expected wait); a large λ means events occur frequently (short expected wait).
Key property: exponential distribution exhibits "memorylessness"—the past doesn't affect future waiting time. If you've already waited 5 minutes, the expected additional wait is still the same as if you just started waiting. This unique property makes it invaluable for modeling processes with no aging effects.
Customers arrive on average every 4 minutes. What's the probability someone arrives within 2 minutes?
There's about a 39% chance a customer will arrive within 2 minutes. Or equivalently, there's a 61% chance you'll have to wait more than 2 minutes.
If you've waited 5 minutes for a bus, the expected additional wait is the same as when you first arrived. Past wait time doesn't influence future waiting. This distinguishes exponential from common intuitions like 'it's been so long, it's due soon'.
λ is inversely related to the mean: higher λ means shorter average wait. If λ = 0.5 (events per hour), mean wait = 1/0.5 = 2 hours. If λ = 2, mean wait = 0.5 hours.
Poisson counts number of events in a time interval; exponential measures time between events. If events follow a Poisson process with rate λ, the time between events is exponential with rate λ. They're two sides of the same coin.
If waiting times aren't exponential, the process may have memory (aging), non-constant rates, or dependencies. Consider Weibull (generalizes exponential), gamma (sum of exponentials), or lognormal distributions instead.
Use P(X ≤ x) to find probability of short waits (things happening soon). Use P(X ≥ x) for tail risk (things taking a long time). Both are useful depending on your question.
No. Exponential is defined only for x ≥ 0 (non-negative values). Negative times don't make sense in real applications. If you input x < 0, the calculator returns no result.
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