{"uid":"cap_4_U1U4uqWue9c1ACaVboj","slug":"probability-distributions-t-distribution-97aedd46","name":"Probability Distributions – t-Distribution","description":"Student's t distribution with df degrees of freedom (> 0). `function` selects the density (pdf), cumulative distribution (cdf) or quantile / inverse-CDF.","url":"https://distribution.openverbs.com/v1/t","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","df"],"properties":{"x":{"type":"number","description":"Evaluation point: the variate for pdf/pmf/cdf, or the probability p ∈ (0,1) for quantile."},"df":{"type":"number","description":"Degrees of freedom ν > 0.","exclusiveMinimum":0},"function":{"enum":["pdf","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_WzjsxboG11zO2vr6c0PTv","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 PDF, CDF, or quantile (inverse CDF) for the Student's t-distribution given a value and degrees of freedom","exampleAgentPrompt":"What is the cumulative probability (CDF) of the t-distribution at x = 1.96 with 30 degrees of freedom?","exampleUseCases":[{"title":"Hypothesis test p-value calculation","prompt":"I have a t-statistic of 2.3 with 15 degrees of freedom — can you give me the CDF value of the t-distribution at that point so I can work out the p-value?"},{"title":"Critical value for confidence interval","prompt":"What is the 97.5th percentile (quantile at p=0.975) of the t-distribution with 29 degrees of freedom? I need this for a 95% confidence interval."},{"title":"Probability density for model verification","prompt":"Can you compute the PDF of the Student's t-distribution at x=0.5 with 5 degrees of freedom so I can verify my stats library is giving the right answer?"}],"resultDescription":"Returns a numerical result representing either the probability density (PDF), cumulative probability (CDF), or the quantile value (inverse CDF) of the Student's t-distribution for the specified evaluation point and degrees of freedom.","failureModes":["Invalid degrees of freedom (df ≤ 0) returns a validation error","For the quantile function, x must be in (0,1); values outside this range return an error","Missing required fields (function, x, df) return a schema validation error","Extremely large or small x values may produce numerical precision issues","Unsupported function enum values return a bad request error"],"whenToPreferThis":"Choose this endpoint when you need precise numerical evaluation of Student's t-distribution functions (PDF, CDF, or quantile/inverse CDF) for statistical computations such as hypothesis testing, confidence interval construction, or Bayesian inference, especially when you need a lightweight, pay-per-call API rather than running a local stats library.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-09-14T18:30:24.222Z","isFirstParty":false}