1. Motor vehicles arrive at a toll gate according to a Poisson process with rate A vehicles per minute. The vehicles arriving at the gate belong to classes 1, 2 with probabilities 1/2, 1/3 or 1/6, respectively. The drivers pay tolls of $1, $2 on depending on the classes 1, 2, or 3 their vehicles belong to. (a) Find the probability that exactly $1 is collected in a period of 2 minutes. (b) Find the probability that the first toll collected is $1. (c) Find the probability that the waiting time between two vehicles that pay $ more than 10 minutes. (d) Find the mean and the variance of the amount in dollars collected in any gi hour. (e) Suppose that 5 vehicles arrived at the toll gate in 10 minutes. Find the pr ability that only $5 dollars are collected and all are collected during the firs minutes