Calculators
Exponential Distribution Calculator
Calculate exponential distribution density, cumulative and upper-tail probabilities, mean, variance and median privately in your browser.
Model the waiting time until the next event.
The exponential distribution models the waiting time until the next event in a process where events occur independently at a constant average rate λ. The density is f(x) = λ × e^(−λx) for x ≥ 0, cumulative probability is P(X ≤ x) = 1 − e^(−λx), and upper-tail probability is P(X > x) = e^(−λx). The mean is 1/λ, the variance is 1/λ², and the median is ln(2)/λ. For λ = 0.25 and x = 4, the density is about 0.09197, cumulative probability is about 0.63212, probability of waiting longer than 4 is about 0.36788, and mean, median and variance are 4, about 2.77259 and 16. Enter rate λ from 0.000000000001 to 1000000000000 and x from 0 to 1000000000000; scientific notation is accepted. The distribution is memoryless, so past waiting time does not change the remaining-time distribution. This is a model of a constant-rate process, not a fit to observed data. Results use floating-point arithmetic rounded to 12 significant digits. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is the memoryless property?
Given that you have already waited longer than s, the chance of waiting at least another t is the same as the chance of waiting t from the start: P(X > s + t | X > s) = P(X > t). The exponential distribution is the only continuous distribution with this property; its discrete counterpart is the geometric distribution.
When is this model suitable?
Use it for waiting times between events that occur independently at a constant average rate, such as time between support tickets arriving or between clicks on a page. Processes with a changing rate, scheduled events, or strong clustering do not follow this model.
Is the density a probability?
No. Density has inverse measurement units and can exceed one, for example with a rate of 5 at x = 0. A single exact value has probability zero in this continuous model; probabilities come from areas under the density, such as P(X ≤ x).
How does it relate to the Poisson distribution?
If inter-arrival times are exponential with rate λ, the number of events in a fixed interval of length t follows a Poisson distribution with mean λ × t. The upper-tail probability e^(−λx) equals the Poisson probability of zero events in an interval of length x.