Problem 4 After receiving their graded exams, students send e-mails to the instructor regarding their grades. E-mails arrive according to a Poisson process with rate d. Each email is then inde- pendently reviewed based on its merit and with probability p, a grade is adjusted. Let T denote the first time a grade is adjusted and let N denote the total number of emails it takes. 1. Calculate P(N = k) for all k e {1, 2, ...}. = 2. Calculate P(N = k|T = t) for all k e {1,2,...} and t > 0. = Problem 5 Passengers arrive at a bus stop as a Poisson process with rate 1. Each passenger is headed towards one of two destinations A and B with probabilities p and 1 p, respectively. Buses to A arrive as a Poisson process with rate pA, while buses to B arrive as a Poisson process with rate us. The arrivals of the buses and passengers and the destination of each passenger are all independent. 1. How many people on average get on the first bus to A? 2. How many people on average get on the first bus to arrive at the stop? 3. How many people on average remain at the stop after the first bus leaves? Problem 4 After receiving their graded exams, students send e-mails to the instructor regarding their grades. E-mails arrive according to a Poisson process with rate d. Each email is then inde- pendently reviewed based on its merit and with probability p, a grade is adjusted. Let T denote the first time a grade is adjusted and let N denote the total number of emails it takes. 1. Calculate P(N = k) for all k e {1, 2, ...}. = 2. Calculate P(N = k|T = t) for all k e {1,2,...} and t > 0. = Problem 5 Passengers arrive at a bus stop as a Poisson process with rate 1. Each passenger is headed towards one of two destinations A and B with probabilities p and 1 p, respectively. Buses to A arrive as a Poisson process with rate pA, while buses to B arrive as a Poisson process with rate us. The arrivals of the buses and passengers and the destination of each passenger are all independent. 1. How many people on average get on the first bus to A? 2. How many people on average get on the first bus to arrive at the stop? 3. How many people on average remain at the stop after the first bus leaves