How to find the marginal density here in the question given?

Question

Let $Y$ have a uniform distribution on the interval $(0, 1)$, and let the conditional density of $X$ given $Y = y$ be uniform on the interval from $0$ to $\sqrt{y}$. What is the marginal density of $X$ for $0 < x < 1$?

Answer 1

Given $$Y=U(0,1)$$ $$f_{X/Y=y}(x|y)=\begin{cases} 1/\sqrt{y}, & 0<x<\sqrt{y}\\ 0, & otherwise \end{cases} $$ Using Conditional probability, $$f_X(x)=f_{X/Y=y}(x|y).f_Y(y)$$ Therefore, $$f_X(x)=\begin{cases} 1/\sqrt{y}, & 0<x<1\\ 0, & otherwise \end{cases}$$