15.3 Exercise 21

A service center on an interstate highway experiences customers in a one-hour period as follows:

• Northbound: Total vehicles: Poisson (200). Twenty percent are trucks.
• Southbound: Total vehicles: Poisson (180). Twenty five percent are trucks.

• Each truck has one or two persons, with respective probabilities 0.7 and 0.3.
• Each car has 1, 2, 3, 4, or 5 persons, with probabilities 0.3, 0.3, 0.2, 0.1, 0.1, respectively

Under the usual independence assumptions, let D be the number of persons to be served. Determine $E \left[D\right]$, $Var \left[D\right]$, and the generating function $g_{D} \left(s\right)$.