This is an elementary problem but my question is regarding the intuition behind plotting the graph of the pdf.
The stick breaking problem (in relation to the topics such as conditional pdf, joint pdf, total and iterated expectations) states that, we have a stick of length $\mathcal{L}$ and we break the stick for the first time at $\mathcal{X}$, which is uniform in $\mathcal{[0,L]}$. We break the stick for second time at $\mathcal{Y}$, which is uniform in $\mathcal{[0,X]}$. Now we need to find the marginal pdf and expectation of $\mathcal{Y}$.
My questions :
$a.$ why in this first diagram we plot $\mathcal{X}$ after $\mathcal{L}$, on the $\mathbb{X}$- axis.
$b.$ In the interval $\mathcal{[L,X)}$ [the thick black line on the $\mathbb{X}$- axis], there is no density, but cannot figure out the reason behind it.
For the question $a.$, here is what I think - $[\mathcal{0,X}] \leq [\mathcal{0,L}]$, therefore while plotting $\mathcal{f_{X}(x), L}$ and $\mathcal{X}$, we should plot $\mathcal{X}$ before $\mathcal{L}$, on the $\mathbb{X}$- axis. But that will be a wrong plot of the marginal pdf of $\mathcal{f_{X}(x)}$ and $\mathcal{X}$, yet I cannot figure out why it would be wrong. Also, understanding $a$ might help me to understand my question $b$.
Calculating marginal pdf and expectation of $\mathcal{Y}$ is not a problem. Please help me to understand the intuition behind plotting the pdf. All helps, explanations are valuable, welcome and appreciated. Also this diagram is taken from the MIT OCW course - Introduction To Probability.
Update : Now it is easy to understand the other pdf plots - 
Your understanding is basically fine; the diagrams are just confusingly labeled.
In the top-right diagram, the $x$ along the bottom axis is just indicating that it's the $x$-axis. There is no particular choice of $x$ in that plot; the point of this plot is to show what $f_X(x)$ would be at different values of $x$.
The other diagrams might be confusing in the same way. For example, the bottom-left plot shows $x$ to the right of $\ell$, but it's not saying $x > \ell$. Instead the $x$ is once again just saying that's the $x$-axis, while the $\ell$ is marking a specific value along the $x$-axis.