a. The joint distribution of the random variables X and Y is given below. X\Y 1 2 3 1 5a 2a 7a 2 9a 6b 8b 3 4a 3b C where a, b and c are unknown constant scalars. i. Given E[X] = 2.5 and the probability of "{X = 1}^{Y = 1} or {X=2}{Y = 2}" is 0.15, determine a, b and c. ii. Calculate the conditional probability for X = 1 given Y = 1. iii. Are the random variables X and Y independent? Justify your answer. b. In an online chat system, there is one staff member answering the queries from customers. A customer will use the chat system for an average of 7.5 minutes according to a Poisson process. The chat system can serve an average of 12 customers per hour according to a second Poisson process. i. What is the probability that a customer will have to wait for being served? ii. Find the average number of customers waiting in the queue and the average waiting time per customer.