Statistical (3x+1) -- problem
  Take an odd number x >0. Then 3x+1 is even and one can find an integer k> 0 so that y= 3x+1/2^k is again odd. We get in this way the mapping T, Tx=y. The paper contains two theorems describing statistical properties of T. The first describes statistical properties of trajectories for large x. The second gives some information about the probability distribution appearing in this problem. 
Yakov Sinai