Given $N$ circles of radius $r$ when $r$ is uniformly distributed in $(r_{min},r_{max})$. These circles are then uniformly distributed inside a big circle of radius $R$, known that $R \gg r$. Assume an isotropic line intersects the big circle, then what is the expected length of the chord that the line created when it crosses the small circles inside the big ones?
In fact, when I fixed $r$, I got the result as \begin{align*} E[l] = \frac{{\pi}r}{2} (*). \end{align*} Then when $r$ is uniformly distributed, we can calculate it as the conditional expectation such as \begin{align*} E[l] & = \int_{r_{min}}^{r_{max}} E[l|r] f_{r}(r) dr \\ & = \int_{r_{min}}^{r_{max}} \frac{E[l|r]}{r_{max} - r_{min}} dr. \end{align*} with $E[l|r] = \frac{{\pi}r}{2}$. However, when I ran the montecarlo simulation to check, the simulation result seemed a little bit different with the analysis ones ( around $10\%$). Could anyone give me a hint about where I am wrong? Thank you very much.
The radius' of the circles within the big circle are randomly changing. So one experiment is like it is depicted below:
Obviously the bigger circles have higher probability to get hit by the red line.
Your calculation is based on the assumption that during an experiment all the circles have the same (random) radius as shown in the next figures:
Then in another experiment the radius of the experiment will be different. Your calculation is correct if the experiments go like in the second case.
However, in the first case the length of the chords are biased upward.
I am not sure if this explanation is enough because I did not do the calculation. But I am pretty sure that this is the problem.