{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"poisson-distribution"},"result":{"entity_type":"calculator","id":"poisson-distribution","calculator_id":"poisson-distribution","canonical_url":"https://tttkmbb.com/statistics/poisson-distribution","name":"Poisson Distribution Calculator","title":"Poisson Distribution Calculator – Probability of k Events at Average Rate λ","category":"statistics","category_name":"Statistics & Probability","tool_name":"calculate_poisson_probability","featured_mcp_tool":false,"description":"Computes the probability of exactly, at most, at least, fewer than or more than k events when events occur independently at a constant average rate λ per interval, plus the mean, variance and standard deviation.","use_when":"You count independent events over a fixed interval of time or space with a known average rate (calls per hour, defects per metre, arrivals per minute) and need the probability of a given count.","do_not_use_when":"The number of trials is fixed and each has a success probability (use binomial-distribution), the rate varies over the interval, or events cluster (over-dispersion).","inputs":[{"name":"lambda","label":"Average rate (λ)","type":"number","required":true,"max":10000,"exclusive_min":0,"description":"Expected number of events in the interval (mean of the distribution).","example":3},{"name":"events","label":"Number of events (k)","type":"integer","required":true,"min":0,"max":100000,"description":"Number of events of interest.","example":2}],"outputs":[{"name":"probability_exact","label":"P(X = k)","type":"number","decimals":6,"description":"Probability of exactly k events."},{"name":"probability_at_most","label":"P(X ≤ k)","type":"number","decimals":6,"description":"Cumulative probability of k or fewer events."},{"name":"probability_at_least","label":"P(X ≥ k)","type":"number","decimals":6,"description":"Probability of k or more events = 1 − P(X ≤ k − 1)."},{"name":"probability_less_than","label":"P(X < k)","type":"number","decimals":6,"description":"Probability of fewer than k events."},{"name":"probability_more_than","label":"P(X > k)","type":"number","decimals":6,"description":"Probability of more than k events = 1 − P(X ≤ k)."},{"name":"mean","label":"Mean","type":"number","decimals":4,"description":"λ."},{"name":"variance","label":"Variance","type":"number","decimals":4,"description":"λ (equal to the mean for a Poisson distribution)."},{"name":"std_dev","label":"Standard deviation","type":"number","decimals":4,"description":"√λ."}],"input_schema":{"type":"object","properties":{"lambda":{"description":"Expected number of events in the interval (mean of the distribution).","type":"number","maximum":10000,"exclusiveMinimum":0,"examples":[3]},"events":{"description":"Number of events of interest.","type":"integer","minimum":0,"maximum":100000,"examples":[2]}},"additionalProperties":false,"required":["lambda","events"]},"output_schema":{"type":"object","properties":{"probability_exact":{"description":"Probability of exactly k events.","type":"number"},"probability_at_most":{"description":"Cumulative probability of k or fewer events.","type":"number"},"probability_at_least":{"description":"Probability of k or more events = 1 − P(X ≤ k − 1).","type":"number"},"probability_less_than":{"description":"Probability of fewer than k events.","type":"number"},"probability_more_than":{"description":"Probability of more than k events = 1 − P(X ≤ k).","type":"number"},"mean":{"description":"λ.","type":"number"},"variance":{"description":"λ (equal to the mean for a Poisson distribution).","type":"number"},"std_dev":{"description":"√λ.","type":"number"}}},"formula":"P(X = k) = e^(−λ) · λ^k / k!; P(X ≤ k) = Σ_{i=0..k} P(X = i); P(X ≥ k) = 1 − P(X ≤ k − 1); mean = variance = λ","method":"Terms are evaluated in log space (log k! accumulated incrementally), so large λ and k do not overflow. Scale λ to the interval you are asking about: 2 calls per minute is λ = 10 for a 5-minute window.","sources":[{"name":"NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.19 Poisson Distribution","url":"https://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm","type":"government","retrieved_at":"2026-09-23"},{"name":"Wikipedia – Poisson distribution","url":"https://en.wikipedia.org/wiki/Poisson_distribution","type":"reference","retrieved_at":"2026-09-23"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"λ = 3, k = 2","inputs":{"lambda":3,"events":2},"expected":{"probability_exact":0.224042,"probability_at_most":0.42319,"probability_at_least":0.800852,"probability_less_than":0.199148,"probability_more_than":0.57681,"mean":3,"variance":3,"std_dev":1.7321},"url":"https://tttkmbb.com/api/v1/calculate/poisson-distribution?lambda=3&events=2"},{"name":"λ = 5, no events","inputs":{"lambda":5,"events":0},"expected":{"probability_exact":0.006738,"probability_at_most":0.006738,"probability_at_least":1,"probability_more_than":0.993262,"std_dev":2.2361},"url":"https://tttkmbb.com/api/v1/calculate/poisson-distribution?lambda=5&events=0"}],"faq":[{"q":"How do I get λ from a rate?","a":"Multiply the rate by the length of the interval in the same units: 4 customers per hour over 15 minutes gives λ = 1."},{"q":"When does Poisson approximate the binomial?","a":"For many trials with a small success probability (n ≥ 20, p ≤ 0.05, or more loosely n·p < 10) the binomial(n, p) is close to Poisson(λ = n·p)."}],"tags":["poisson distribution","poisson probability","events per interval","rate lambda","arrival probability"],"related":[{"calculator_id":"binomial-distribution","reason":"Exact probabilities for a fixed number of trials."},{"calculator_id":"probability-of-events","reason":"Combine probabilities of two events."}],"links":{"html":"https://tttkmbb.com/statistics/poisson-distribution","markdown":"https://tttkmbb.com/statistics/poisson-distribution.md","json":"https://tttkmbb.com/statistics/poisson-distribution.json","api":"https://tttkmbb.com/api/v1/calculate/poisson-distribution","schema":"https://tttkmbb.com/api/v1/calculators/poisson-distribution","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:28:22Z"}