A class of n students takes a test consisting of m questions. Suppose that student i submitted answers to the first ml questions. a. The grader randomly picks one answer, call it (I, J), where I is the student ID 1, 1. . . . , n) and J is the question number (taking values . . . , m). Assume that all answers are equally likely to be picked. Calculate the number (taking values 1, joint and the marginal PMFs of I and J.
b. Assume that an answer to question j. if submitted by student i, is correct with probability Plj. Each answer gets a points if it is correct and gets b points otherwise. Calculate the expected value of the score of student i.