Exponential Distribution Calculator

How to Use the Calculator

1. Open Specify parameter by and choose whether you want to enter the rate \(\lambda\) (events per time) or the mean waiting time \(\mu\). Only the matching field is shown.
2. If you chose the rate, fill in Rate (λ) and pick the time unit it is expressed in (per second, per minute, per hour, per day, per year). If you chose the mean, fill in Mean waiting time (μ) with its unit instead; the calculator converts it via \(\lambda = 1/\mu\).
3. Enter Time (t) — the waiting time you are interested in — and select its unit under "in".
4. Optionally fill in Interval end (b) to also get the probability \(P(t \le X \le b)\). Leave it empty to skip this result.
5. Read the results: density \(f(t)\), cumulative probability \(P(X \le t)\), survival probability \(P(X > t)\), the interval probability, the mean waiting time, and a summary line with the median, the 90th-percentile wait and a plain-language interpretation.

What Is the Exponential Distribution?

The exponential distribution describes the waiting time until the next event in a process where events happen independently at a constant average rate — a Poisson process. Typical applications include the time between customer arrivals, the time until a radioactive particle decays, the lifetime of an electronic component, or the time until the next server failure.

The distribution is defined by a single parameter, the rate \(\lambda\). The probability density \(f(t) = \lambda e^{-\lambda t}\) is largest at \(t = 0\) and decays smoothly, meaning short waits are the most likely. The cumulative function \(F(t) = 1 - e^{-\lambda t}\) gives the probability that the wait ends at or before time \(t\); its complement, the survival function \(S(t) = e^{-\lambda t}\), gives the probability of still waiting after \(t\).

The mean and the standard deviation are both \(1/\lambda\), but the median \(\ln(2)/\lambda \approx 0.693/\lambda\) is smaller. This tells you something practical: more than half of all waits end before the average. A large wait is possible but increasingly rare — for example, only 5% of waits exceed \(3/\lambda\).

A defining property of the exponential distribution is memorylessness: \(P(X > s + t \mid X > s) = P(X > t)\). If a light bulb has an exponentially distributed lifetime and has already burned for a year, its chances of surviving another year are exactly the same as when it was new. This makes the distribution the continuous analogue of the geometric distribution and the natural companion of the Poisson distribution, which describes the number of events in a fixed period rather than the time between them.

Assumptions, Limitations and Tips

• Constant rate. The model assumes events occur at a steady average rate at all times. If the rate varies (e.g., rush-hour arrivals, wear-out of mechanical parts), the exponential distribution will under- or overestimate the probabilities.
• Independence. One event must not influence the timing of the next. Clustering of events violates this assumption.
• Memorylessness. If the process truly is exponential, already-elapsed waiting time carries no information about the remaining wait. Real lifetimes of machines and humans usually are not memoryless; for those, reliability models such as the Weibull distribution fit better.
• Units. The calculator converts all units internally, so you can enter the rate per hour and the time in minutes without doing any arithmetic yourself. One year is taken as 365 days.
• Density vs. probability. \(f(t)\) is a density per time unit, not a probability — it can exceed 1. The meaningful probability statements are \(F(t)\), \(S(t)\) and the interval probability.
• If you need to model counts of events rather than waiting times, use the Poisson distribution calculator; for yes/no outcomes over a fixed number of trials, see the binomial distribution calculator, and for symmetric continuous data the normal distribution calculator.