Let x=gpowery;p, then when p≠0 y=gpower−1x=px+11/p−px+1−1/p2 and it is easy to see that ∂y∂x=∂gpower−1x∂x=px+11/p+px+1−1/p2(px+1) for p>0, x>−1/p and the derivative is positive, which means that gpowery;p is a monotonically increasing function. Now, X=gpowerY is considered as a function of a random variable Y∼GPN(μ,σ,p). The density for X is obtained as follows: fXx=∂∂xPgpowerY;p≤x=fYpx+11/p−px+1−1/p2px+11/p+px+1−1/p2(px+1)=1K2πσ2exp−12σ2x−μX)2,−1/p<x<∞ that is precisely the density of a TNμX,σ2,−1/p,∞ random variable. For p<0, it can be proved analogously that fXx=1K2πσ2I−∞,−1/pxexp−12σ2x−μX2,