Weibull Distribution Calculator — PDF, CDF & Reliability

Formula

How this is calculated

The Weibull distribution is defined by two parameters: shape k (sometimes written β) and scale λ (sometimes η, the characteristic life). The shape parameter controls how the failure or event rate changes over time. When k < 1 the failure rate decreases — this models infant mortality or early-life failures. When k = 1 the distribution reduces to an exponential, meaning a constant failure rate. When k > 1 the failure rate increases with time — typical of wear-out failures in mechanical parts. The scale λ is the value at which the CDF equals 1 − e⁻¹ ≈ 63.2%, regardless of shape.

The probability density function f(x) gives the relative likelihood at a specific value. Integrating it from 0 to x gives the cumulative distribution function F(x) = 1 − exp(−(x/λ)^k), which is the probability the variable is less than x. The survival function P(X > x) = 1 − F(x) is widely used in reliability engineering to express the probability of surviving past a given time. All statistics — mean, median, mode, variance — are derived from the gamma function Γ, approximated here using the Lanczos algorithm.

Limitations: the two-parameter Weibull has no location shift (it starts at 0); for datasets that start above zero, a three-parameter Weibull is needed. The gamma approximation introduces tiny numerical error (< 10⁻¹⁵ for most k). Very small k near zero can produce very large PDF values near the origin.