(1) Let f : [n] r [n] be a permutation. Axed point of f is an element x E [n] such that f(x) = x. Now consider random permutations of [n] and let X be the random variable which represents the number of xed points of a given permutation. {a} What is the probability that X = 0? {b} What is the probability thatX = n 2? (c) What is the probability that X = n 1? {d} What is the expectation of X? (Hint: As usual, express X as the sum of wellchosen indicator variables and use the linearity of expectation.) (e) Let tk be the number of permutations of [n] that have exactly I: fixed points. Prove that 23:\" ktk = n!. (Hint: Use parts {aHd) of the problem and consider two different expressions for E 00.) O A. It is a discrete random variable. O B. It is a continuous random variable. O C. It is not a random variable. d. Is the time it takes to fly from City A to City B a discrete random variable, a continuous random variable, or not a random variable? O A. It is a continuous random variable. O B. It is a discrete random variable. O C. It is not a random variable. a. Is the number of hils to a website in a day a discrete random variable, a continuous random variable, or not a random variable? O A. It is a continuous random variable. O B. It is a dacrete random variable. O G. his not a random variable. I. Is the time required to download a filo from the Internet a discrete random variable, a continuous random variable, or not a random variable? O A. I is a discrete random variable. O B. It is a continuous random variable O C. It is not a random variable. Click to select your answer. Sirie for LMC MacBook Air BST Q W E R T U caps lock A S D F G H Z X C V B N control option command