Inverse of the CDF of a Gamma RV

restart

with(Statistics):

C := RandomVariable(GammaDistribution(2, 2))

f := unapply(CDF(C, t), t) assuming t > 0;

plot(f(t), t=0..10);

# f being a bijection from [0, +infinity) to [0, 1], its inverce does exist

s := solve({f(t)=u, t >=0}, t) assuming u >= 0, u <= 1;

pc := plot(CDF(C, t), t=0..10, color=blue):
T  := plottools:-transform((x, y) -> [y, x]):
plots:-display(T(pc), plot(rhs(op(s[1])), u=0..0.95, color=red), plot(rhs(op(s[2])), u=0..0.95, color=cyan, transparency=0.5, thickness=5));