How can you calculate the probability distribution of the period length of a linear congruential generator? That is $X_{n+1} = (aX_n + c) \bmod m$ where $a$ is chosen uniformly at random from $\{1,\dots, m-1\}$ and $c$ is chosen uniformly at random from $\{0,\dots, m-1\}$ and $m$ is a fixed prime. Take $X_0$ to be some arbitrary value from $\{0,\dots, m-1\}$.
If it is hard to do exactly, is it possible to give good bounds for the cdf?
There's not much of a distribution there. The period is $1$ if $a=1$ and $c=0$ or if $a\ne1$ and $X_0=c/(1-a)$; otherwise it's $m-1$.