{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"exponential-distribution"},"result":{"entity_type":"calculator","id":"exponential-distribution","calculator_id":"exponential-distribution","canonical_url":"https://tttkmbb.com/statistics/exponential-distribution","name":"Exponential Distribution Calculator","title":"Exponential Distribution Calculator – Waiting-Time Probabilities from a Rate λ or Mean","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_exponential_distribution_probability","featured_mcp_tool":false,"description":"Computes the density, the cumulative probability P(X ≤ x), the survival probability P(X > x) and optionally P(x < X ≤ x2) for an exponential distribution given its rate λ or its mean 1/λ, plus mean, median, variance and standard deviation.","use_when":"You model the time between independent random events at a constant rate (time to failure, time between arrivals, radioactive decay) and need the probability of waiting at most or more than a given time.","do_not_use_when":"You need the probability of a number of events in an interval (use poisson-distribution), the trials are discrete (use geometric-distribution), or the failure rate changes over time (Weibull).","inputs":[{"name":"rate","label":"Rate (λ)","type":"number","required":false,"exclusive_min":0,"description":"Events per unit time. 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It is the continuous counterpart of the geometric distribution and the inter-arrival time of a Poisson process with rate λ.","sources":[{"name":"NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.7 Exponential Distribution","url":"https://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm","type":"government","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Exponential distribution","url":"https://en.wikipedia.org/wiki/Exponential_distribution","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. 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If both are given they must agree (rate × mean = 1)."},{"q":"Why is the median less than the mean?","a":"The distribution is right-skewed: many short waits and a few very long ones. The median is ln 2 ≈ 0.693 times the mean."}],"tags":["exponential distribution","waiting time probability","time between events","memoryless","mean time between failures"],"related":[{"calculator_id":"poisson-distribution","reason":"Number of events in an interval at the same rate."},{"calculator_id":"half-life","reason":"Exponential decay expressed as half-life."},{"calculator_id":"geometric-distribution","reason":"Discrete-trial analogue."}],"links":{"html":"https://tttkmbb.com/statistics/exponential-distribution","markdown":"https://tttkmbb.com/statistics/exponential-distribution.md","json":"https://tttkmbb.com/statistics/exponential-distribution.json","api":"https://tttkmbb.com/api/v1/calculate/exponential-distribution","schema":"https://tttkmbb.com/api/v1/calculators/exponential-distribution","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:44:52Z"}