Show transcribed image text Problem 9 Suppose that n students enter a final exam and sit down at a desk with an exam. The professor has chosen a seating chart at random and labeled the exams exams at each desk, but forgot to tell the students. Let X be the number of people who sit at their own exam 1. Let Ak be the event that person k sits at their own exam. Prove or dis- prove: The events A, and Ak are independent for j k. 2. Find E(X). Hint: Write X as a sum of random variables 3. Find Var(X). Hint: Write X as a sum, and compute covariances 4. Find exactly P(X ã€‰ 0) = P (UR=1 Ak) 5. Given an approximation to P (X ã€‰ 0) for large.

Problem 9 Suppose that n students enter a final exam and sit down at a desk with an exam. The professor has chosen a seating chart at random and labeled the exams exams at each desk, but forgot to tell the students. Let X be the number of people who sit at their own exam 1. Let Ak be the event that person k sits at their own exam. Prove or dis- prove: The events A, and Ak are independent for j k. 2. Find E(X). Hint: Write X as a sum of random variables 3. Find Var(X). Hint: Write X as a sum, and compute covariances 4. Find exactly P(X ã€‰ 0) = P (UR=1 Ak) 5. Given an approximation to P (X ã€‰ 0) for large.