Your friend’s professor gives out reasonably hard exams 70% ofthe time, and ridicu- lously hard exams 30% of the time. On hardexams, each student’s score on the exam is a normally distributedrandom variable with ?H = 70 and ?H = 10. On ridiculously hardexams, each student’s score on the exam is a normally distributedrandom variable with ?R = 50 and ?R = 15. Suppose you have fourfriends in the class, not just one. Let A be the average score ofyour four friends: A= (F1 +F2 +F3 +F4)/ 4 Where F1 is your firstfriend’s score, and F2 is your second friend’s score. Find E[A] andV ar(A) if the exam is ridiculously hard. (e) Find E[A] and V ar(A)if the exam is reasonably hard. (f) Since A is the sum of normalrandom variables, it is itself a normal random variable. Find P (A> 65) if the exam is reasonably hard, and if it is ridiculouslyhard. (g) If A is greater than 65, what is the probability the examwas ridiculously hard?
