Consider a person who commutes to the city from a suburb by train. He is accustomed to leaving his home between 7:30 and 8:00 A.M. The drive to the railroad station takes between 20 and 30 minutes. Assume that the departure time and length of trip are independent random variables, each uniformly distributed over their respective intervals. There are 3 trains that he can take, which leave the station and arrive in the city precisely on time. The first train leaves at 8:05 A.M. and arrives at 8:40 A.M., the second leaves at 8:25 A.M. and arrives at 8:55 A.M., the third leaves at 9:00 A.M. and arrives at 9:43 A.M.
(i) Find the mean and variance of his time of arrival in the city.
(ii) Find the mean and variance of his time of arrival under the assumption that he leaves his home between 7:30 and 7:55 A.M.