We introduce the free analogue of the classical beta prime distribution by the multiplicative free convolution of the free Poisson and the reciprocal of free Poisson distributions, and related free analogues of the classical F, T, and beta distributions. We show the rationales of our free analogues via the score functions and the potentials. We calculate the moments of the free beta prime distribution explicitly in combinatorial fashion by using non-crossing linked partitions and demonstrate that the free beta prime distribution belongs to the class of the free negative binomials in the free Meixner family.

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我们通过自由泊松和自由泊松分布的倒数的乘法自由卷积引入经典 beta 素数分布的自由类似物，以及经典 F、T 和 beta 分布的相关自由类似物。我们通过得分函数和势来展示我们的自由类似物的基本原理。我们通过使用非交叉链接分区以组合方式显式计算自由 beta 素数分布的矩，并证明自由 beta 素数分布属于自由 Meixner 族中的自由负二项式类别。