A miner has been trapped in a mine with three doors. One takes him to freedom after one hour, one brings him back to the mine after 3 hours and the third one brings him back after 5 hours. Suppose that he wishes to get out of the mine and that he does so by choosing one of the three doors uniformly at random and continues to do so until he is free. Find the generating function, the mean and the variance for the time it takes him to reach freedom.