Suppose we have $X_i \stackrel{\text{i.i.d}}{\sim} U(0,1)$ and we define a stopping time $N =\min \{n \mid X_1+\dots +X_n >1\}.$ How can one find the distribution of $X_N$?
I have seen how to find the distribution of $N$ using the renewal function $m(t)$ (in particular, $E[N]=e$) but I cannot figure out how to see the distribution of $X_N$. Through bootstrap methods, I see it is not just uniform, but I don't see why.