15.3 Exercise 16

A junction point in a network has two incoming lines and two outgoing lines. The number of incoming messages N₁ on line one in one hour is Poisson (50); on line 2 the number is $N_{2} \sim$ Poisson (45). On incoming line 1 the messages have probability $p_{1 a} = 0 . 33$ of leaving on outgoing line a and $1 - p_{1 a}$ of leaving on line b. The messages coming in on line 2 have probability $p_{2 a} = 0 . 47$ of leaving on line a. Under the usual independence assumptions, what is the distribution of outgoing messages on line a? What are the probabilities of at least 30, 35, 40 outgoing messages on line a?