Exponential Distribution Calculator
Waiting time between events at rate λ: CDF and survival via expm1, the median beside the mean, and the memorylessness card that only this distribution can honestly print.
Exponential Distribution Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Constant hazard only — aging/wear-out processes need Weibull (boundary stated)
- Single rate; no piecewise or mixture models
In short: A server fails at rate λ = 0.5 per year: P(fail within 2 years) = 1 − e^−1 = 0.632121, the median lifetime is ln 2/λ = 1.386294 years, the mean is 1/λ = 2 and the variance 1/λ² = 4. The card only the exponential can print: memorylessness — P(X > 4 | X > 2) = P(X > 2) = 0.367879 exactly, because the survivor ratio e^−λ(s+t)/e^−λs telescopes to e^−λt. Having waited changes nothing: the machine does not age in this model.
Formula
P(X ≤ t) = 1 − e^−λt · median = ln 2/λ · mean = 1/λ, variance = 1/λ²
The CDF is computed as −expm1(−λt) — the one numerically honest form at both small and large λt; the naive 1 − exp(−λt) loses digits exactly when t is small.
Worked Example
- Give the rate λ (per unit time) and the time t.
- Read P(X ≤ t) and the survival P(X > t) beside it.
- Compare median (ln 2/λ) with mean (1/λ) — the long right tail pulls the mean past the median.
- Read the memorylessness card: the conditional survival on your own numbers.
λ = 0.5, t = 2: P(X ≤ 2) = 0.632121, survival 0.367879, median 1.386294, mean 2, variance 4, and P(X > 4 | X > 2) = P(X > 2) = 0.367879 exactly.
Strengths & Limits Of This Model
Where this engine is strong
- expm1-based CDF that keeps digits at both extremes
- Memorylessness computed live — the identity the page is famous for
Where it stops
- No Weibull/gamma generalizations
- No censored-data fitting — you bring λ, the page does not estimate it
Practical Use Cases
Reliability
P(failure before warranty ends)
Ops
P(next ticket inside an hour)
Queues
inter-arrival gaps for a Poisson process
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Exponential Distribution Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What does “memoryless” actually claim?
That the future depends only on now: P(X > s + t | X > s) = P(X > t). Survive three years and your remaining-life curve is identical to a brand new unit’s — the survivor ratio e^−λ(s+t)/e^−λs telescopes to e^−λt. The card computes it live on your numbers; the identity is exact, not an approximation.
Why is the median smaller than the mean?
The tail. Half of all waits finish before ln 2/λ = 0.6931/λ, but the few very long waits drag the mean out to 1/λ — on the defaults 1.386294 versus 2. Quoting the mean alone flatters “typical”; this page prints both.
What is the hazard rate here?
Flat: λ per unit time, forever. That is the memorylessness restated — no aging, no wear-in, no infant mortality. Real machines and real patients have bathtub-shaped hazards; when your hazard bends, Weibull is the honest upgrade, and this page says so as a boundary.
How does this pair with the Poisson page?
Same process, two views. Poisson counts events per window; exponential times the gaps between events — same λ. Tickets at 2.5 per hour mean waits are exponential at rate 2.5, and the count of gaps in an hour is Poisson at 2.5. The pages cross-link because together they are one model.
Why compute the CDF as −expm1(−λt)?
Digits. At small λt the naive 1 − exp(−λt) subtracts two nearly equal numbers and sheds most of its precision; expm1 is built to survive exactly that regime, and it degenerates gracefully to 1 when λt is enormous. The page never prints a probability whose digits it did not earn.
My data shows wear-out — still exponential?
No: wear-out means the hazard RISES with age, and the whole point of this model is a flat hazard. Fitting λ to lifetime data with aging gives predictions that are safest exactly where you need them least. The boundary note names Weibull; this page would rather refuse the premise than fake the curve.
What if I know the mean instead of the rate?
Invert it: λ = 1/mean. A mean wait of 20 minutes is λ = 0.05 per minute. Every formula here runs on the rate — type the reciprocal and the cards read the same either way, median ln2/0.05 = 13.862943 minutes included.
Can the exponential ever be negative time?
No — t below 0 gets P = 0, stated plainly: waiting time cannot be negative, and the model’s support starts at 0. The refusal is the model’s own boundary, printed rather than laundered through a formula that was never asked.