Inference for the Weibull Distribution

For β = 1 the Weibull distribution coincides with the exponential distribution with mean α. In general, α represents the .632-quantile of the Weibull distribution regardless of the value of β since Fα,β(α) = 1 − exp(−1) ≈ .632 for all β > 0. Figure 1 shows a representative collection of Weibull densities. Note that the spread of the Weibull distributions around α gets smaller as β increases. The reason for this will become clearer later when we discuss the log-transform of Weibull random variables. The m moment of the Weibull distribution is E(X) = αΓ(1 +m/β)