A new family of univariate probability distributions called the T − R {Y} power series family of probability distributions is introduced in this paper by compounding the T − R {Y} family of distributions and the power series family of discrete distributions. A treatment of the general mathematical properties of the new family is carried out and some sub-families of the new family are specified to depict the broadness of the new family. The maximum likelihood method of parameter estimation is suggested for the estimation of the parameters of the new family of distributions. A special member of the new family called the Gumbel–Weibull–{logistic}–Poisson (GUWELOP) distribution is defined and found to exhibit both unimodal and bimodal shapes. The GUWELOG distribution is further applied to a real multi-modal data set to buttress its applicability.

中文翻译：

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概率分布的 T-R {Y} 幂级数族

本文通过复合 T − R {Y} 分布族和离散分布的幂级数族，引入了一个新的单变量概率分布族，称为概率分布的 T − R {Y} 幂级数族。对新族的一般数学性质进行了处理，并指定了新族的一些子族来描述新族的广泛性。对于新分布族的参数估计，建议采用参数估计的最大似然法。定义了称为 Gumbel-Weibull-{logistic}-Poisson (GUWELOP) 分布的新家族的一个特殊成员，并发现它表现出单峰和双峰形状。GUWELOG 分布进一步应用于真实的多模态数据集以支持其适用性。