2. (32 points) You are interested in the average score on Statistics final exams. The final exam has two types of questions, each with a different probability a student will get the question correctly: 3 easy questions with probability of getting the question correct of 0.85. Each easy question is worth 2 points. 2 hard questions with probability of getting the question correct of 0.5. Each hard question is worth 3 points. Every question is independent. There are 50 students in the class, and you are interested in the average score on the exam, X. Let X; denote student i's exam grade. The average score on the exam is defined as follows: 50 1 X Xi. 50 (a) (16 points) Find: The expectation of an individual student's score, E[X] = i The variance of an individual student's score, Var(Xi) = xi Note that X is defined as follows: 3 2 Xi X = 2 Eij + 3 Hij, j=1 j=1 where Eij is 1 if student i gets the jth easy question correct (0 otherwise) and Hi; likewise is 1 if student i gets the jth hard question correct (0 otherwise). (b) (8 points) What is the standard deviation of average scores, x? (c) (8 points) What is the probability that the average score is below 3.5?