Is there a family of probability distributions for ...?

Question

Is there a family of probability distributions for $P(x|y,n)=\frac{\Gamma(n+y)}{\Gamma(n)\Gamma(y)}(x+1)^{-n-y}x^{y-1}$ ?
$n>0$ and $y>0$
Has it an explicit expression for the CDF?

Answer 1

Off the top of my head, I can't think of a name, but I wouldn't be surprised if there's a connection with the Beta family of distributions.

For your second question, $$ F(x) = \dfrac{\Gamma(n+y)}{\Gamma(n)\Gamma(y)} \dfrac{x^y}{y}[ _2F_1(y,n+y;1+y;-x)] $$ where $$_2F_1(a,b;c;z)=\sum_{k\ge0}a_k \dfrac{b_k}{c_k} \dfrac{z^k}{k!}.$$

Answer 2

This is the Beta-prime distribution (there is probably more than you need to know at the end of the link, including the CDF).