Let $U$ be $~U [0,1] $and let $Y = U^{\frac{1}{2}}$ Let $U$ be $\sim \mathcal{U}[0,1]$ and let $Y = U^{1/2}$.
I'm having trouble finding the $E(Y)$. How do I go about doing this? 

Answer using CDF transformation method:
$$F_{Y}(y) = P(\sqrt(U) <y)$$
$$F_{Y}(y)=P(U<y^2) = \frac{y^2-0}{1-0} = y^2 ; 0<y<1$$
$$f_{Y}(y) = \frac{dF_{Y}(y)}{dy} = 2y; 0<y<1$$
$$E(Y) = \int_{0}^{1} y\cdot 2y dy$$
$$ = \frac{2y^3}{3}|_{0}^{1}$$
$$ = \frac{2}{3}$$
Thanks
Satish
