{
  "slug": "weibull-distribution",
  "term": "Weibull distribution",
  "aliases": [
    "Weibull lifetime model",
    "Two-parameter Weibull"
  ],
  "definition": "A Weibull distribution is a continuous probability model whose two-parameter form uses positive shape and scale parameters to describe nonnegative event times.",
  "note": "With shape k > 0 and scale eta > 0, S(t) = exp(-(t/eta)^k) for t >= 0, and h(t) = (k/eta)(t/eta)^(k-1) for t > 0. Eta uses the time unit of the modeled variable; k is dimensionless. Shape below, equal to or above one gives decreasing, constant or increasing hazard. The k = 1 case is exponential. [nist-weibull]\n\nAt t = eta, the cumulative event probability is 1 - exp(-1), approximately 63.2%, regardless of shape. This follows from the CDF; eta is not generally the mean or median. A single two-parameter Weibull has monotone hazard, so cannot by itself represent a decrease followed by an increase. These are model properties, not measured drive behavior. [nist-weibull]\n\nSpecify the event and clock before interpreting a fitted shape. FAST 2007 fitted HPC1 intervals between cluster disk replacements, not individual disk lifetimes. A marginal interval fit does not establish independence between consecutive events; the paper separately examined correlation. Do not transfer those fitted parameters into a device-aging model. [schroeder-2007]\n\nConditioning on survival to age a gives P(T <= a + u | T > a) = 1 - S(a + u)/S(a), for a >= 0 and u >= 0. This model deduction differs from the unconditional probability by age a + u; the latter includes events before a. For Weibull k = 1, the conditional probability is independent of age. [nist-weibull]",
  "status": "shipping",
  "advantages": [],
  "limitations": [],
  "related": [
    "hazard-rate",
    "survival-function",
    "interarrival-time",
    "afr",
    "mean-residual-life"
  ],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "citations": [
    {
      "id": "nist-weibull",
      "title": "NIST e-Handbook 8.1.6.2: Weibull, formulas and parameter conventions",
      "url": "https://www.itl.nist.gov/div898/handbook/apr/section1/apr162.htm",
      "accessed": "2026-11-05"
    },
    {
      "id": "schroeder-2007",
      "title": "Schroeder and Gibson: Disk failures in the real world, FAST 2007, sections 2.4, 5.2 and 5.3",
      "url": "https://static.usenix.org/events/fast07/tech/schroeder/schroeder_html/index.html",
      "accessed": "2026-11-05"
    }
  ],
  "updated": "2026-11-05"
}