At time t = 0, you start getting mail messages according to a Poisson process of rate . Each e-mail is independently spam with probability p, an invitation to a party with probability q, or neither with probability 1 p q. Each spam message is immediately deleted, each party invitation is immediately directed to a folder labeled IMPORTANT, and other messages immediately go to a folder named OTHER. We can assume that the process of getting emails goes on indefinitely. Starting with empty folders at t = 0, if 10 messages get collected in the IMPORTANT folder before 10 messages get collected in the OTHER folder, you will conclude that you are "very popular". Otherwise you conclude that you are not very popular. (a) Find the CDF of T , the time at which you decide whether or not you are "very popular". (Your answer can contain summation(s) of terms.) (b) How many of the first 19 non-spam messages must be party invitations for you to con- clude that you are very popular? (c) What is the probability that you end up deciding you are very popular? (Hint: consider the ongoing process of e-mails and use your answer to the previous part.)