Transformation of Uniform(0,1) random variable

Question

Let $X \sim U[0,1]$. Find the pdf of $Y=4\sqrt{X}(1-\sqrt{X})$.

I have been studying transformation of random variables and came across this exercise.

Can anyone provide me a hint on how to proceed? Thanks.

Answer 1

One possible way is to compute the CDF and hen differentiate with respect to $y$.

Here's a start, try to complete it.

Let $y \in (0,1)$,

\begin{align}Pr(Y \le y)&= Pr(-4X+4\sqrt{X}-y \le 0)\\ &=Pr\left(\sqrt{X} \le \frac{1-\sqrt{1-y}}{2}\right) + Pr\left(\sqrt{X} \ge \frac{1+\sqrt{1-y}}{2}\right)\\ &=\left( \frac{1-\sqrt{1-y}}{2}\right)^2 +1- \left( \frac{1+\sqrt{1-y}}{2}\right)^2 \end{align}