This exponential distribution calculator models waiting times between random events — time until the next customer, call, failure or decay — giving probabilities, percentiles, mean and median from the rate λ.

P(X ≤ x) — by time x–
P(X > x) — still waiting–
Mean / Median–
Percentile value–

The exponential is memoryless: having waited 10 minutes already, your expected additional wait is still the full mean — the process doesn’t “owe” you an event. It’s the continuous twin of the Poisson counts on our Poisson Distribution Calculator: Poisson counts events per interval, exponential times the gaps between them, same λ.

P(X ≤ x) = 1 − e^(−λx) — One Parameter, Whole Story

The exponential is the unique continuous memoryless distribution: mean = 1/λ, median = ln2/λ ≈ 0.693/λ, and the survival curve is a pure decaying exponential — the same function as radioactive decay on our Half-Life Calculator, because decay gaps are its native habitat. Part of our Statistics Calculators collection.

Worked Example (Verified)

  • λ = 0.5 per hour, x = 2 h (defaults): P(wait ≤ 2 h) = 1 − e^−1 = 0.6321; still waiting = 0.3679; mean 2.0 h, median 1.386 h (median < mean — the right skew's signature); 90th percentile wait = 4.605 h
  • Call-center version: calls at λ = 12/hour (mean gap 5 min): probability the next minute is call-free = e^−0.2 ≈ 0.819 — staffing math in one exponent
  • The memoryless trap as a number: survived 2 h already (prob 0.368), the chance of needing 2 MORE hours is again 0.368 — elapsed time purchases nothing
  • Reliability: components with λ = 0.001/hour have mean life 1,000 h but median 693 h, and 36.8% fail before the "mean lifetime" — why warranties are set from percentiles, not means

Where the Model Fits (and Doesn't)

  • Fits: radioactive decay, inter-arrival times of truly random events, service gaps, time-to-failure for memoryless (constant-hazard) electronics in useful life.
  • Doesn't fit: anything that ages (hazard rising — Weibull territory), schedules (buses bunch deterministically), or queues with memory.
  • SD equals the mean (both 1/λ) — a quick diagnostic: if your waiting-time data's SD ≈ mean, exponential is plausible; if not, look elsewhere.
  • Percentile formula inverts the CDF: x_q = −ln(1−q)/λ — used here for service-level promises ("95% answered within…").

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Frequently Asked Questions

How do you calculate exponential distribution probabilities?

P(X ≤ x) = 1 − e^(−λx), where λ is the event rate (1/mean). For λ = 0.5 and x = 2: 1 − e^−1 = 0.6321. The survival probability P(X > x) is just e^(−λx). Mean = 1/λ, median = 0.693/λ.

What does memoryless actually mean?

P(X > s + t | X > s) = P(X > t): past waiting doesn't change future odds. A component that has run 500 hours has the same failure outlook as a fresh one (if truly exponential). It's the defining property — shared among continuous distributions by the exponential alone.

How are exponential and Poisson distributions related?

Same process, two questions: if events occur at rate λ, the count in a fixed window is Poisson(λt), and the gap between consecutive events is Exponential(λ). Measure one and you've parameterized the other — λ here is identical to the Poisson page's rate.

What is λ if I only know the average wait?

λ = 1 ÷ mean wait. Average gap of 5 minutes → λ = 0.2 per minute (or 12 per hour — keep units consistent; this calculator's "mean waiting time" field converts for you).

Why is the median less than the mean?

The distribution is right-skewed: most waits are short, a long tail of rare long waits drags the mean up. Median/mean = ln 2 ≈ 0.693 always — a fixed fingerprint of the exponential family, and the reason "average wait 10 minutes" still leaves most people served sooner.

Is this exponential distribution calculator free?

Yes — free, instant, no sign-up.