Finding new probability density function with change of variable Y=sqrt(X)

Question

Say we have a given distribution, such as X~No(a, b). I am trying to find the pdf and mean for $Y=\sqrt{X}$.

I know the steps for finding the PDF, but since Y can only take on positive values, then the new PDF is only valid for Y>0.

Then how does this affect the mean, and how would I go about finding the mean?

Answer 1

If $Y=\sqrt{|X|}$, then, for every $y\gt0$, $$ f_Y(y)=2y\cdot(f_X(y^2)+f(-y^2)).$$