⏱️ Exponential Distribution Calculator
Model the time between random, independent events with the exponential distribution.
What the exponential distribution models
The exponential distribution describes the time or distance between independent random events that occur at a constant average rate, such as the time between customer arrivals, equipment failures, or radioactive decay events. It is defined by a single rate parameter, lambda (λ), representing the average number of events per unit of time.
How this calculator works
Enter the rate λ and a value x (an elapsed time or distance). The calculator finds the probability density f(x) = λe-λx, the cumulative probability P(X ≤ x) = 1 – e-λx (the chance the event happens by time x), and P(X > x), the chance it takes longer than x. It also reports the mean (1/λ) and variance (1/λ²) of the distribution.
The exponential distribution assumes events happen independently at a constant average rate (a Poisson process) and is "memoryless", meaning the probability of waiting a bit longer does not depend on how long you have already waited. This tool is for general statistics education and planning.