Beta prime distribution with optional power and scale parameters
BetaPrimeDistribution[p, q]
BetaPrimeDistribution[p, q, β]
BetaPrimeDistribution[p, q, α, β]
Mean[BetaPrimeDistribution[p, q]]
→ Piecewise[{{p/(-1 + q), q > 1}}, Infinity]PDF[BetaPrimeDistribution[p, q], x]
→ Piecewise[{{(x^(-1 + p)*(1 + x)^(-p - q))/Beta[p, q], x > 0}}, 0]CDF[BetaPrimeDistribution[p, q], x]
→ Piecewise[{{BetaRegularized[x/(1 + x), p, q], x > 0}}, 0]