Travelers arrive at a bus station from 6 a.m., according to a Poisson process with rate λ = 1 per minute. The first bus leaves T minutes after 6 a.m. (a) Calculate the mean and the variance of the...

Travelers arrive at a bus station from 6 a.m., according to a Poisson process with rate λ = 1 per minute. The first bus leaves
T
minutes after 6 a.m.

(a) Calculate the mean and the variance of the number of travelers ready to board this bus if (i)
T
has an exponential distribution with mean equal to 15 minutes and (ii)
T
is uniformly distributed between 0 and 20 minutes.

(b) Calculate the average number of passengers on the bus if it leaves at 6:15 and if its capacity is 20 passengers.