Exponential Distribution Calculator
The exponential distribution describes the time you wait for the next event when events happen at a steady average rate lambda. Enter the rate and an evaluation point x. The calculator returns the probability density at x, the cumulative probability that the wait is at most x, and the distribution's mean and variance. It is the standard model for the time between arrivals in a Poisson process, equipment lifetimes with a constant hazard rate, and radioactive decay intervals. Every output comes from the exact closed-form expression, with no approximation.
Exponential distribution formulas
f(x) = lambda * e^(-lambda * x) for x >= 0, else 0
F(x) = 1 - e^(-lambda * x) for x >= 0, else 0
Mean = 1 / lambda
Variance = 1 / lambda^2
The density starts at lambda when x is 0 and decays exponentially. The cumulative function rises from 0 toward 1 as x increases, giving the probability that the event has occurred by time x.
Worked example and notes
- With lambda = 0.5, the mean wait is 1/0.5 = 2 time units.
- At x = 2 the cumulative probability is 1 minus e^(-1), about 0.6321.
- The density at x = 2 is 0.5 times e^(-1), about 0.1839.
- The variance is 1/0.25 = 4, so the standard deviation equals the mean of 2.
- The distribution has no memory: past waiting time never changes future odds.
Exponential distribution: frequently asked questions
What is the exponential distribution?
The exponential distribution models the waiting time until the next event in a process where events occur at a constant average rate lambda and independently of one another. It is the continuous analogue of the geometric distribution and is the unique continuous distribution with the memoryless property.
What is the density and cumulative function?
For x at or above 0, the density is lambda times e to the power minus lambda x, and the cumulative distribution function is 1 minus e to the power minus lambda x. Both are 0 for negative x. The cumulative value gives the probability that the wait is at most x.
What are the mean and variance?
The mean waiting time is 1 divided by lambda, and the variance is 1 divided by lambda squared. So the standard deviation equals the mean, a hallmark of the exponential distribution.
What does memoryless mean here?
Memoryless means the probability of waiting an additional time t does not depend on how long you have already waited. Formally P(X greater than s plus t given X greater than s) equals P(X greater than t). The exponential is the only continuous distribution with this property.
What are valid inputs?
The rate lambda must be strictly positive. The evaluation point x can be any real number; for negative x the density and cumulative value are both 0, since the distribution is supported on the non-negative reals.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Exponential distribution.
- NIST Digital Library of Mathematical Functions: Special functions reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.