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Exponential Distribution Calculator
The exponential distribution models the time until the next event when events occur continuously at a constant average rate λ: the time between customer arrivals, the lifetime of a component with a constant failure rate, or the gap between radioactive decays. Enter the rate, choose the probability type and the time value(s), and the calculator evaluates F(x) = 1 − e⁻ᵏˣ and the required difference.
The steps show the exponent, the value of e⁻ᵏˣ and the final probability, with a shaded density curve. The mean 1/λ, standard deviation 1/λ and median ln 2/λ are given, along with the density λe⁻ᵏˣ at x. If you know the mean waiting time instead of the rate, enter λ = 1 ÷ mean.
The exponential distribution is memoryless: the chance of waiting another t units does not depend on how long you have already waited. It is the continuous counterpart of the geometric distribution, and the number of events in a fixed period follows the Poisson distribution with the same rate.
Result and step-by-step solution
Mean 1/λ = 2, SD = 2, median ln2/λ = 1.3863
- Formulaf(x) = λe−λx, F(x) = P(X ≤ x) = 1 − e−λx
- Evaluate the CDFF(3) = 1 − e−0.5 × 3 = 1 − 0.22313 = 0.77687
- Apply the ruleP(X ≤ 3) = 0.77687
Density at x = 3: f(x) = 0.11157. The exponential distribution is memoryless: P(X > s + t | X > s) = P(X > t).
Frequently asked questions
- What is λ and how does it relate to the mean?
- λ is the average number of events per unit time. The mean waiting time is 1/λ, so a rate of 0.5 per minute means a 2-minute average wait.
- How do I find the probability of waiting more than x?
- P(X > x) = e⁻ᵏˣ. Choose P(X > x) in the calculator and the steps show the exponent.
- What is the median of an exponential distribution?
- ln 2 ÷ λ ≈ 0.693/λ, which is less than the mean because the distribution is right-skewed.
- Does SciPy use the rate or the scale?
- SciPy’s expon uses scale = 1/λ. This calculator takes the rate λ directly, so enter the reciprocal of a scale parameter.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.