{"uid":"cap_E1pb-zepnPMTh5Lq3J2MP","slug":"poisson-distribution-calculator-8778589e","name":"Poisson Distribution Calculator","description":"Poisson distribution with rate `lambda` (> 0, ≤ 1e9). `function` selects the mass (pmf), cumulative distribution (cdf) or quantile / inverse-CDF.","url":"https://distribution.openverbs.com/v1/poisson","method":"POST","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method","bodyType","body"],"properties":{"body":{"type":"object","required":["function","x","lambda"],"properties":{"x":{"type":"number","description":"Count k (integer for pmf) for pmf/cdf, or the probability p ∈ (0,1) for quantile."},"lambda":{"type":"number","description":"Rate λ > 0.","exclusiveMinimum":0},"function":{"enum":["pmf","cdf","quantile"],"type":"string"}},"additionalProperties":false},"type":{"type":"string","const":"http"},"method":{"enum":["POST"],"type":"string"},"bodyType":{"enum":["json","form-data","text"],"type":"string"}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.004","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.004/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.004","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.004","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_ly1XMmhahPP5cYspfOwvB","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.004","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes PMF, CDF, or quantile (inverse CDF) for the Poisson distribution given a rate parameter λ and a count or probability value.","exampleAgentPrompt":"What is the probability of exactly 3 events occurring in a Poisson process with a rate of 2.5 events per interval? Use the PMF function.","exampleUseCases":[{"title":"Rare event probability in operations","prompt":"We expect on average 4 customer complaints per day. What is the probability we get exactly 6 complaints tomorrow? Use the Poisson PMF with lambda 4 and k 6."},{"title":"Cumulative arrival rate analysis","prompt":"If a server receives on average 10 requests per second, what is the probability of receiving 12 or fewer requests in any given second? Compute the Poisson CDF with lambda 10 and x 12."},{"title":"Quantile threshold for capacity planning","prompt":"Our call center handles Poisson-distributed calls with an average rate of 8 per hour. What call volume corresponds to the 99th percentile so we can plan staffing? Use the Poisson quantile function with lambda 8 and p 0.99."}],"resultDescription":"Returns a single numeric value: either the probability mass (PMF), the cumulative probability (CDF), or the smallest integer k such that CDF(k) ≥ p (quantile), for the specified Poisson distribution parameters.","failureModes":["Invalid λ ≤ 0 returns a validation error","Probability p outside (0,1) for quantile function returns an error","Non-integer or negative k for PMF/CDF may return an error or unexpected result","Missing required fields (function, x, lambda) returns a schema validation error","Payment failure via x402 protocol prevents access"],"whenToPreferThis":"Choose this endpoint when you need fast, on-demand computation of Poisson distribution probabilities or quantiles without setting up a statistics library or runtime environment. Ideal for agents performing probabilistic reasoning about count-based events (arrivals, failures, defects) in real time. Prefer this over a generic math API when you specifically need Poisson PMF, CDF, or inverse CDF evaluated server-side with a simple HTTP call.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-09-14T12:34:22.267Z","isFirstParty":false}