Collatz stopping-time and Poisson distribution, and connection to other problems? I read many threads about Collatz here - so don't worry, this is no attempt to any proof, just asking about a curious fact:
This graph gives the stopping-time of Collatz sequences up to $n=10^8$

(source: http://en.wikipedia.org/wiki/File:CollatzStatistic100million.png ) and it's distribution looks very similar to a Poisson distribution.
Is there some known reason why the Collatz-sequence stopping time behaves like a poissonian distribution?
What are the connections to other mathematical problems: Does the Collatz-conjecture imply other conjectures, or do other conjectures imply the Collatz-conjecture?
Thank you!
 A: As to the observed distribution of total stopping times for integers $n \leq 10^8$,
I think heuristically this can be explained quite well by the obvious stochastic model
(multiply $n$ by $3/2$ or $1/2$, each with probability $1/2$, repeat this
until the number gets $\leq 1$ and count the number of steps this takes).
For literature on such stochastic considerations, see Lagarias'
annotated bibliography on the conjecture (http://arxiv.org/abs/math/0309224).
Proving anything is of course quite a different task!
The Collatz conjecture can be formulated in quite a number of equivalent
ways, see also Lagarias' bibliography. Though so far the conjecture is by far
not as well-embedded into known parts of mathematics as for example the Riemann
Hypothesis. Namely I am not aware of, say, important 'theorems' which are
proved up to the Collatz conjecture. Nevertheless, I think the fact that an
easy-to-state question like the Collatz conjecture seems so intractable
suggests that certain important things are not understood so far.
Hence I don't think the conjecture will seem as isolated as now for all
future -- but of course this is my personal view.
