In this work, we have taken up some distributions, mostly Weibull family, whose quantile functions could not be obtained using the traditional inversion method. We have solved the same quantile functions by using the inversion method only, with the additional help of transcendental functions like the Lambert W function. The usage of the Lambert W function has helped derive closed-form solutions for mathematical models in many fields, for which explicit or exact solutions were not available and approximation was the only known way of approaching it.

翻译：本研究针对若干分布（主要为威布尔族分布）展开讨论，这些分布的量函数无法通过传统反演方法求得。我们仅借助反演方法，并辅以兰伯特W函数等超越函数，成功求解了相同的量函数。兰伯特W函数的运用为多个领域的数学模型推导出闭式解提供了帮助，对于这些模型而言，此前因无法获得显式或精确解，近似处理曾是唯一可行的研究途径。