Unlock: Weibull Distribution

The Weibull distribution is the parametric workmate of reliability and survival analysis. Two parameters: a shape k and a scale lambda. The hazard rate is monotone in time, increasing when k > 1 (wear-out), constant when k = 1 (Exponential, memoryless), decreasing when k < 1 (early-failure / infant-mortality). Mean is lambda Gamma(1 + 1/k); variance is lambda squared times (Gamma(1 + 2/k) minus Gamma(1 + 1/k) squared). MLE for k has no closed form and is solved by Newton-Raphson on the profile score equation. Applications: component lifetimes, time to event in clinical trials, wind speeds, extreme-value Type III.