For the record, to answer the question in the OP: The desired PDF $P_n(r)$ of the distance $r$ from the origin for arbitrary $n$ can be expressed as an integral over Bessel functions $J_0$,
$$P_n(r)=r\int_0^\infty k [J_0(k)]^n J_0(k r)\,dk,\;\;0<r<n,$$
see this <A HREF="https://math.stackexchange.com/a/612395/87355">MSE posting</A> for a derivation.