I have a ML problem with dependent y-variables that are skew-symmetric to the right. I perform the transformation $\sqrt y$ on my dependent variable before running the ML algorithm, and then make a set of predictions on my test data. These are transformed predictions, and I would like to transform them back from $\sqrt y$ to $y$. However, from my statistics class I know that:
$$E[y^\frac 12]^2 \neq E[y] $$
what we do know, based on the assumption from the transformation, is that:
$$y^\frac 12 \sim N(\mu,\sigma^2) $$
However I'm not sure how to use this to get back to $E[y]$. Also, I think it is important to note that each prediction $y_i$ corresponds to the predictions for how well individual person i performs in a game, and that each person i doesn't necessarily have the same mean and variance, since some individuals are better than others / more variable at the game.
Any thoughts on this will help!