The number of intercity shipment orders arriving daily at a transportation company is a random variable X with the following probability distribution:
a. Verify that P(x) is a probability distribution.
b. Find the cumulative probability function of X.
c. Use the cumulative probability function computed in (b) to find the probability that anywhere from one to four shipment orders will arrive on a given day.
d. When more than three orders arrive on a given day, the company incurs additional costs due to the need to hire extra drivers and loaders. What is the probability that extra costs will be incurred on a given day?
e. Assuming that the numbers of orders arriving on different days are independent of each other, what is the probability that no orders will be received over a period of five working days?
f. Again assuming independence of orders on different days, what is the probability that extra costs will be incurred two days in a row?