3. Suppose your rst midterm raw exam score is a random variable with mean 70 and variance 25. This question is hypothetical and does not necessarily reect reality. a) The professor will curve the exam and everyone will receive a new curved midterm score. Suppose that the curve is such that your new curved score is New Score = it Score + 20. This means that a student with a raw score of 80 would get a curved grade of (E - 80) + 20 = 84. What is the mean of the new curved scores? b] Now, assume that the curve is such that your new curved score is New Score = Score + 10. What is the mean of the new curved scores? c) Suppose that you know that each of the rst 2 problem sets has a mean of 40 and a variance of 16. Each problem set has a maximum of 50 points. Your professor needs to report to the Dean's office your mid-semester score, which is simply a weighed average of your 1'51 midterm score and your scores on each of the 2 problem sets, where the lEl midterm constitutes NZ of your mid-semester score while the each problem sets is worth U4 of your mid-semester grade. Suppose that the l"1 midterm score and each problem set are independent. What is the mean of your mid-semester score? d) Suppose that your mid-semester grade was determined only by your 1\" midterm grade and one of the two problem sets. Assume that the midterm and problem set grades are independent and that the mid-semester grade is a simple average of the two. What is the expected value for your mid-semester grade