Let a random variable $X$ be uniform distribution in $(0,1)$ and another variable $Y=X^2$. I calculate the sample space of $Y$ is also $(0,1)$, and the cdf and pdf of $Y$ are $F_Y(y)=\sqrt{y}$ and $f_Y(y) = \frac{1}{2\sqrt{y}}$. I think $Y$ should also be a random variable and the integration of pdf of $Y$ should give 1 but it does not seems right. I don't expect the following integration gives me 1. $ \int^1_0{\frac{1}{2\sqrt{y}}}dy = -\frac1 4 y^{-\frac 3 2}|^1_0 $
Did I calculate the sample space and pdf and the integration correctly? Or a variable derived from a random variable is not necessarily a random variable.