Okay if I have a distribution that the only thing I am sure of is that it is Gaussian/Normal and over time(actually a long time) ends up as a Gamma Function for which I know the parameters. Is there a way to determine what transform would have occurred to the Normal Function to arrive at the known Gamma Function and normal Or is it not going to occur because the power of exp are just not compatible. I know I can calculated the Coxbox transform to go from Gamma to Normal and could then look at it the inverse of the transform it gives me, but is there a better way to determine a transform that would go from Normal to known Gamma.

Same question but with johnson distribution rather than Gamma

Thanks in anticipation.