Exponential Distribution Calculator
Compute exponential distribution probabilities, mean, and variance from a rate parameter λ and a value x.
Probability type
Probability for the selected region
- 1
−λ × x
−0.5 × 2 = -1 - 2
e^(−λx)
e^(-1) = 0.367879 - 3
P(X < x) = 1 − e^(−λx)
1 − 0.367879 = 0.6321Cumulative probability.
How does this calculator work?
For an exponential distribution with rate λ, the chance an event occurs before x is P(X<x) = 1 − e^(−λx) and after x is P(X>x) = e^(−λx). The mean equals 1/λ and the variance equals 1/λ². Enter λ and x to get these probabilities plus the density curve.
Formula
How this is calculated
The exponential distribution models the waiting time between independent events that occur at a constant average rate λ. Enter the rate λ (events per unit of time or space) and a value x at which to evaluate the distribution. The rate must be positive and x must be non-negative.
The probability density function is f(x) = λe^(−λx) for x ≥ 0. The cumulative probability P(X < x) = 1 − e^(−λx) gives the chance the event happens before x, while the survival probability P(X > x) = e^(−λx) gives the chance you wait longer than x. The dropdown selects which of these is shown as the headline result; both are also listed in the breakdown.
The distribution has mean 1/λ and variance 1/λ², so larger rates mean shorter expected waits and tighter spread. It is memoryless: P(X > s + t | X > s) = P(X > t), so elapsed waiting time does not change the future distribution. Edge cases: λ ≤ 0 or x < 0 are invalid and return no result; as x grows, P(X < x) approaches 1 and P(X > x) approaches 0.
Frequently asked questions
λ is the average number of events per unit of time (or distance). Its reciprocal 1/λ is the mean waiting time between events.
Because the probability of waiting an additional time t does not depend on how long you have already waited: P(X > s + t | X > s) = P(X > t).
Yes. If you know the mean waiting time m, set λ = 1/m, since the mean of an exponential distribution equals 1/λ.
Also known as
TG we-Calculate Editorial Team. (2026). Exponential Distribution Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/exponential-distribution-calculator
TG we-Calculate Editorial Team. "Exponential Distribution Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/exponential-distribution-calculator.
TG we-Calculate Editorial Team, "Exponential Distribution Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/exponential-distribution-calculator
@misc{wecalculate_exponential_distribution_calculator, title = {Exponential Distribution Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/exponential-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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