Outcomes of the random variable \(Z\) represent the number of customers that are waiting in a queue to be serviced at Fast Lube, a quick stop automobile lubrication business, when the business opens at 9 A.M. on any given Saturday. The probability model \((R(Z), f(z))\) for the random variable \(Z\) is given by:

\(f(z)= \begin{cases}.5^{z+1} \text { for } z & \in R(Z)=\{0,1,2,3, \ldots\} \\ 0 & \text { elsewhere }\end{cases}\)

a. Derive the cumulative distribution function for \(Z\).

b. What is the probability that there will be less than 10 people waiting?

c. What is the probability that there will be more than 3 people waiting?

d. Given that no more than two people will be waiting, what is the probability that there will be no customers when business opens at 9 A.M.?