If $x$ is a uniformly random number in $[0,1]$, what distribution should the $n$-th term in its continued fraction expansion follow?
What is the expected vale of $a_n$ in $[a_0;a_1,a_2,\dots]$?
Here is the expansion for $\pi$.
What does it say about a number if there is some regularity in the sequence? Why does $e$ have regularity, but $\pi$ apparently does not?