Other Distributions Calculator (binomial/Poisson/t/chi²) mechanism

Model scope

The listed binomial and Poisson PMFs are standard and correct under their stated constraints. The overall specification needs review because Student's t and chi-squared PDFs, the numeric integration algorithm, convergence tolerance, tail treatment, and plotting semantics are not specified. The existing honest-label warning is appropriate, but it does not resolve the reproducibility and validation ambiguities.

• The binomial PMF formula is correct, but the required constraints are omitted: n must be a nonnegative integer, k must be an integer with 0 <= k <= n, and 0 <= p <= 1.
• The Poisson PMF formula is correct, but the required constraints are omitted: lambda must satisfy lambda >= 0 and k must be a nonnegative integer. The lambda = 0, k = 0 case requires the convention 0^0 = 1 for the PMF limit.
• The Student's t and chi-squared density formulas are not provided, so the claimed PDF calculations cannot be independently verified from the specification.
• Numeric density integration is underspecified: no integration algorithm, finite-bound transformation, tail handling, singularity handling, quadrature step or tolerance is given. Therefore CDF results are not deterministic or reproducible.
• For chi-squared distributions, the domain x >= 0 must be enforced; the PDF is zero for x < 0 and may have an endpoint singularity when degrees of freedom are less than 2.
• For Student's t and chi-squared distributions, degrees of freedom must satisfy nu > 0.

Units and conventions

• Binomial n and k are dimensionless integer counts; p is a unitless probability in [0,1].
• Poisson lambda is a dimensionless expected count for the stated observation interval; k is a dimensionless nonnegative integer count. If lambda is supplied as a rate, the interval duration must also be supplied and lambda = rate multiplied by duration.
• Student's t x is dimensionless, normally a standardized statistic; degrees of freedom nu is dimensionless and positive.
• Chi-squared x is dimensionless and nonnegative; degrees of freedom nu is dimensionless and positive.
• Reported PDF values have reciprocal units of x in a general continuous-variable interpretation, although these distributions conventionally use dimensionless standardized variables.
• CDF values and PMF values are dimensionless probabilities.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "distribution": "binomial",
    "n": 4,
    "p": 0.5,
    "k": 2,
    "plot_sample_k": [
      0,
      2,
      4
    ]
  },
  "expected_outputs": {
    "pmf": 0.375,
    "cdf": 0.6875,
    "mean": 2,
    "variance": 1,
    "plot_sample_points": [
      {
        "k": 0,
        "pmf": 0.0625
      },
      {
        "k": 2,
        "pmf": 0.375
      },
      {
        "k": 4,
        "pmf": 0.0625
      }
    ],
    "key_curve_features": {
      "support": "k in {0,1,2,3,4}; PMF is zero outside this support",
      "mode": "k=2",
      "symmetry": "PMF is symmetric about k=2"
    }
  },
  "working": "PMF at k=2: C(4,2) * 0.5^2 * (1-0.5)^(4-2) = 6 * 0.25 * 0.25 = 0.375. CDF through k=2: P(0)+P(1)+P(2) = [1*0.5^4] + [4*0.5^3*0.5] + [6*0.5^2*0.5^2] = 0.0625 + 0.25 + 0.375 = 0.6875. Mean = n*p = 4*0.5 = 2. Variance = n*p*(1-p) = 4*0.5*0.5 = 1. Plot samples: P(0)=C(4,0)*0.5^4=0.0625; P(2)=0.375; P(4)=C(4,4)*0.5^4=0.0625."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The degrees-of-freedom input ν=0 produces multiple undefined operations in the Student's t density. The calculator must reject it rather than return a numeric PDF; accepting it without validation would be a failure. The discrete CDF convention also materially changes the result unless explicitly specified.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json