Motor vehicles arrive at atoll gate according to a Poisson process with rate

=2vehicles per minute. The vehicles arriving at the gate belong toclasses1, 2or3

with probabilities1/2, 1/3or1/6, respectively. The drivers pay tolls of$1, $2or$5

depending on theclasses1, 2,or3 their vehicles belong to.

1. Find the probability that exactly $1 is collected in a period of 2 minutes.
2. Find the probability that the first toll collected is $1.
3. Find the probability that the waiting time between two vehicles that pay $5 is more than 10 minutes.
4. Find the mean and the variance of the amount in dollars collected in any given hour.