Weibull Distribution Calculator - PDF, CDF & Reliability

The Weibull distribution is a continuous probability distribution named after the Swedish engineer and mathematician Waloddi Weibull, who used it in 1951 to model material strength and fatigue. It is now one of the most important distributions in reliability engineering, survival analysis, wind-speed modeling, and extreme-value theory because its shape parameter k allows it to model increasing, constant, or decreasing failure rates — three very different physical behaviours — within a single flexible family. The distribution is defined by two parameters. The shape parameter k (sometimes written β) controls whether the failure rate increases, decreases, or remains constant over time. When k > 1, the failure rate increases with time — this models wear-out failures typical of mechanical components, where parts degrade with use. When k = 1, the Weibull distribution reduces exactly to the exponential distribution with constant failure rate, modelling purely random failures such as electronic components failing at a steady background rate. When k < 1, the failure rate decreases over time — this models infant-mortality failures, where defective items fail early and the survivors become more reliable. The scale parameter λ (sometimes written η) is the characteristic life: at x = λ, the CDF equals 1 − e⁻¹ ≈ 63.2%, regardless of k. The probability density function (PDF) f(x) gives the relative likelihood of observing a failure at exactly time x. The cumulative distribution function (CDF) F(x) gives the probability that a component will have failed by time x — this is also called the unreliability. The reliability function R(x) = 1 − F(x) gives the probability of survival past time x, which is the primary metric for warranty and maintenance planning. The hazard rate h(x) = f(x) / R(x) is the instantaneous failure rate at time x given survival up to that point; in engineering it is called the force of mortality or hazard function. The mean of the Weibull distribution is λ · Γ(1 + 1/k), where Γ is the gamma function. The median is λ · (ln 2)^(1/k). The mode (most likely failure time) is λ · ((k−1)/k)^(1/k) when k > 1 and zero when k ≤ 1. The variance is λ² · [Γ(1 + 2/k) − (Γ(1 + 1/k))²]. Weibull analysis appears in fleet maintenance scheduling, aircraft component certification, wind-energy resource assessment, earthquake-return-period estimation, and cancer survival studies. This calculator performs all standard Weibull computations in a single step, using the Lanczos approximation for the gamma function to maintain high numerical accuracy across a wide range of parameter values.