Question 2. (55 marks) Answer the following questions. A professor visits his lab on any day with probability 0.5. A student working with the professor visits the lab on any day with probability 0.8. Let NA be the number of days the professor visits the lab over a week (7 days). Let NS be the number of days the student visits the lab over a week. Further let N be the number of days in a week that they both meet. Assume that they both meet on days both visit the lab. (a) (15 marks) Derive the joint probability mass function of N A and N . (b) (5 marks) Using the denition of independent random variables show whether NA and N are independent or not. (c) (15 marks) Calculate the expectation E[NA|N = n] and use this to further calculate E[NA]. (d) (20 marks) Suppose we are interested in those weeks in which the professor visited the lab more number of days than the student. Derive the conditional expectation of N A. [Hintz First write down the given event in terms of the described random variables. Calculate the appropriate conditional marginal PME]