On a recent trip to the SC DMV, I asked an employee to estimate what percentage of SC drivers arrive to renew their driver's license with one that...

3. On a recent trip to the SC DMV, I asked an employee to estimate what percentage of SC drivers arrive to renew their driver's license with one that is currently expired. She responded that about 30 percent of all such renewals were of this type.

(a) Suppose you observe Y , the number of SC DMV customers seeking renewal to find the first one with an expired license. What is the distribution of Y ? Plot the pmf and cdf of Y side by side (like in the notes). Place an "×" on the pmf indicating where E(Y ) is.

(b) Let W denote the number of SC DMV customers seeking renewal to find the 3rd one with an expired license. What is the distribution of W ? Plot the pmf and cdf of W side by side . Place an "×" on the pmf indicating where E(W) is.

(c) Obviously, in parts (a) and (b), you are assuming that Bernoulli trial assumptions hold. State what these are in this application (e.g., think of each customer seeking renewal as a "trial.")

(d) In parts (a) and (b), find the probability that

among the first 6 customers seeking renewal, none have expired licenses.

you have to observe 10 or more customers seeking renewal to find the 3rd one with an expired license.

Note: For practice, I want you to do part (d) "by hand," that is, show all the calcula- tions with pencil and paper. Use R to check your work.