  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 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.
