Compute PDF, CDF, percentiles, and reliability statistics for the exponential distribution with interactive rate parameter and visual density curve.
The exponential distribution calculator computes probabilities, percentiles, and reliability statistics for the exponential distribution — the continuous counterpart of the Poisson process. It models the time between independent events that occur at a constant average rate λ.
Common applications include time until the next customer arrives, lifetime of electronic components, wait times at service counters, and intervals between radioactive decays. The exponential distribution is uniquely characterized by the memoryless property: the probability of waiting another t units is the same regardless of how long you've already waited.
This tool provides point PDF/CDF evaluation, interval probabilities, a visual density curve, percentile table, and a reliability analysis table showing survival probability over time.
Use the preset examples to load common values instantly, or type in custom inputs to see results in real time. The output updates as you type, making it practical to compare different scenarios without resetting the page. This supports fast what-if analysis for reliability and waiting-time decisions.
The exponential distribution is fundamental in reliability engineering, queuing theory, survival analysis, and telecommunications. Any scenario involving "time until next event" with a constant rate leads to exponential modeling.
This calculator provides all essential calculations in one place — from basic probabilities to reliability analysis — making it invaluable for engineering coursework and professional applications.
PDF: f(x) = λe^(−λx) for x ≥ 0. CDF: F(x) = 1 − e^(−λx). Mean = 1/λ. Variance = 1/λ². Median = ln(2)/λ. Survival: S(x) = e^(−λx).
Result: P(X ≤ 2) = 0.8647 (86.47%)
With λ = 1 (one event per unit time on average), the probability of waiting at most 2 time units is 1 − e^(−1×2) = 1 − 0.1353 ≈ 86.5%.
In reliability engineering, the exponential distribution models components with a constant failure rate — the "flat" portion of the bathtub curve. The survival function S(t) = e^(−λt) gives the probability a component survives past time t. The hazard rate h(t) = λ is constant, meaning the instantaneous failure rate doesn't depend on age.
The M/M/1 queue assumes exponential inter-arrival and service times. Key results: average wait time = 1/(μ − λ), where μ is service rate and λ is arrival rate. The system is stable only when λ < μ (arrival rate less than service rate).
The exponential is Gamma(1, λ), Weibull with shape 1, and a special case of the beta distribution of the second kind. The minimum of n independent Exponential(λᵢ) variables is Exponential(Σλᵢ), making it ideal for parallel system reliability.
If a light bulb has survived 100 hours, the probability it lasts another 50 hours is the same as a brand new bulb lasting 50 hours. Only the exponential (and geometric) distributions have this property.
They're reciprocals. If customers arrive at rate λ = 3 per hour, the average time between arrivals is 1/3 hour (20 minutes).
Use exponential when the failure rate is constant (random failures). Use Weibull when failure rate changes over time — increasing (wear-out) or decreasing (burn-in).
If events follow a Poisson process (λ events per unit time), then the time between events is exponentially distributed with the same λ. Use this as a practical reminder before finalizing the result.
Not well — human mortality rate increases with age, violating the constant rate assumption. The Weibull or Gompertz distributions are better for human lifetimes.
The maximum likelihood estimate is λ̂ = n / Σxᵢ (number of observations divided by total observed time). Keep this note short and outcome-focused for reuse.