Let $X$ and $Y$ be continuous random variables with probability density functions $P_{X}(x)$ and $P_{Y}(y)$, respectively. Further, let $Y=X^{2}$ and
$$P_{X}(x)=\begin{cases}1, & x \in(0,1] \\ 0, & \text { otherwise }\end{cases}$$
Which one of the following options is correct?
- $P_{Y}(y)=\begin{cases} \frac{1}{2 \sqrt{y}}, & y \in(0,1] \\ 0, & \text { otherwise }\end{cases}$
- $P_{Y}(y)=\begin{cases}1, & y \in(0,1] \\ 0, & \text { otherwise }\end{cases}$
- $P_{Y}(y)=\begin{cases} 1.5 \sqrt{y}, & y \in(0,1] \\ 0, & \text { otherwise }\end{cases}$
- $P_{Y}(y)=\begin{cases} 2 y, & y \in(0,1] \\ 0, & \text { otherwise }\end{cases}$
