i.i.d. Weibull Random Variables

The main purpose in this discussion is to find the distribution function of the sum of independent and identical distributed Weibull random variables. The Weibull distribution is a two-parameter distribution which by adjusting a scale parameter (denoted by c) and a shape parameter (denoted by β) we can obtain a variety of shapes to fit experimental data. Thus, this distribution is highly adaptable and widely used in practice. Without loss of generality, we can assume the scale parameter to be one. In addition, the Weibull distribution is widely used in reliability modeling since other distributions such as exponential (c=1), Rayleigh (c=2), and normal (if a suitable value for the shape parameter is chosen) are special cases of the Weibull distribution. In the version of hazard-rate function, when c>1, the hazard rate is a monotonically increasing function with no upper bound that describes the wear-out region, and when c=1, the hazard rate becomes constant (constant failure-rate region), and when c<1, the hazard rate is a monotonically decreasing with time (the early failure-rate region). This enables the Weibull distribution to describe the failure rate of many failure data in many kinds of application