14.2 Exercise 28

A shop which works past closing time to complete jobs on hand tends to speed up service on any job received during the last hour before closing. Suppose the arrival time of a job in hours before closing time is a random variable $T \sim$ uniform $\left[0 , 1\right]$. Service time Y for a unit received in that period is conditionally exponential $\beta \left(2 - u\right)$, given $T = u$. Determine the distribution function for Y.