Exponential Distribution Calculator

The exponential distribution models the time between consecutive independent events that occur at a constant average rate. It is the continuous counterpart of the geometric distribution and is widely used in reliability analysis, queuing theory, and survival analysis. A defining property is that it is memoryless — the probability of an event occurring in the next interval is always the same, regardless of how much time has already passed.

λ (lambda) is the rate parameter representing the average number of events per unit of time. For example, if buses arrive at a stop at an average of 3 per hour, then λ = 3. The mean time between events is 1/λ, so a higher λ means events happen more frequently and the mean waiting time is shorter.

The probability density function (PDF) is f(x) = λe^(−λx) for x ≥ 0. The cumulative distribution function (CDF), giving P(X ≤ x), is F(x) = 1 − e^(−λx). The survival function P(X > x) = e^(−λx). The mean is μ = 1/λ, the variance is σ² = 1/λ², and the median is ln(2)/λ.

Use the CDF formula: P(X < x) = 1 − e^(−λx). For example, with λ = 0.8 and x = 2.7, P(X < 2.7) = 1 − e^(−0.8 × 2.7) ≈ 0.8847. This tells you there is approximately an 88.47% chance the event occurs before time 2.7.

Use P(x1 < X < x2) = F(x2) − F(x1) = e^(−λx1) − e^(−λx2). Enter the lower bound x1 and upper bound x2 in the optional interval fields above, and the calculator will compute this automatically for you.

Memoryless means that the probability of waiting an additional time t for an event is the same regardless of how long you have already waited. Formally, P(X > s + t | X > s) = P(X > t). For example, if a machine has been running for 100 hours without failing, the probability it fails in the next hour is identical to when it was brand new.

Common examples include the time between arrivals of customers at a service counter, the lifespan of electronic components, the time until the next earthquake, radioactive decay intervals, and the time between phone calls at a call center. Any scenario where events occur independently at a constant average rate can be modeled with the exponential distribution.

The Poisson distribution counts the number of events occurring in a fixed time interval (a discrete distribution), while the exponential distribution models the continuous time between those events. They are closely linked: if the number of events per unit time follows a Poisson distribution with rate λ, then the waiting time between events follows an exponential distribution with the same rate λ.