Weibull

Abstract

The Weibull distribution was formally introduced by Waloddi Weibull, a Swedish mathematician in 1939. The distribution was earlier used by a Frenchman, Maurice Frechet in 1927, and applied by R. Rosin and E. Rammler in 1933. The Weibull distribution has shapes that range from exponential-like to normal-like, and the random variable, w, takes on values of γ or larger. A related distribution, the standard Weibull with variable, x, has values of zero or larger. Both distributions have the same parameters (k₁, k₂) and these form the shape of the distribution. When k₁ ≤ 1, the mode of the standard Weibull is zero and the shape is exponential-like; when k₁ > 1, the mode is larger than zero, and when k₁ is 3 or larger, the shape is normal–like. The mathematical equations for the probability density and the cumulative probability are shown and are easy to compute. However, the calculation of the mean and variance of the distribution are not so easy to compute and require use of the gamma function. Methods to estimate the parameters, γ, k₁, k₂, are described when sample data is available. When no data is available, and an expert type person provides approximations of some measure of the distribution, methods are shown how to estimate the parameter values.