PMF of two uniformly distributed breakpoints

Question

Question: A stick of length $L>0$ is broken at a uniformly random point X. Given that $X=x$, another breakpoint $Y$ is chosen uniformly on the interval $[0, x]$. Find the joint PDF of $X$ and $Y$.

What I got:

1. I know that $X \sim Unif(0, L)$, so $P(X=x) = \frac{1}{L}$.
2. Furthermore, we know that $Y|X=x \sim Unif(0, x)$. Therefore, $P(Y=y | X=x) = \frac{1}{x}$.
3. Using Baym's rule I get $$P(Y=y, X=x) = P(Y=y | X=x) \cdot P(X=x) = \frac{1}{x \cdot L}$$ for $x \in [y, L]$ and $y\in [0,x]$ and zero otherwise. [I edited the support]

The marginal distribution of $Y$ is therefore given by \begin{align} P(Y=y) &\equiv \int_y^Ldx \;P(Y=y, X=x) \\ &= \int_y^Ldx \;\frac{1}{x \cdot L} \\ &=\frac{\ln{(L)} - \ln{(y)}}{L} \end{align}