May I ask if my approach is valid? In attempting this I arrive at:

$\displaystyle f_{X}(x) = \frac{1}{b-a} = \frac{1}{1-0}$

Then finding the cdf of $\displaystyle Y=U^{1/2}$ I have:

$\displaystyle F_{y}(y) = P(Y \leq y) = P(U^{1/2} \leq y) = \int_{0}^{y^2} f_{X}(x)\ dx = y^2$ for $\displaystyle 0 \leq y \leq 1$

So my cdf is

$\displaystyle F_{Y}(y) =\begin{cases}0 & y<0 \\ y^2 & 0<y \leq 1\\1 & y>1 \\ \end{cases}

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Then the pdf is the derivative of this

$\displaystyle f_{Y}(y) =\begin{cases} 0 & y<0 \\ 2y & 0<y \leq 1 \\0 & y>1 \\ \end{cases}

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