Show that the ratio X/N also has a binomial distribution

        questions based on chapter 2 exercises: (1) based on exercise 22 A) Discuss how you might map correlation values from the interval [-1,1] to the interval [0,1]. B) Consider two time series as two sets on numbers: e.g. S1 = {3, 5, 7, 8 , .., 4, 7} and S2 = {5, 5, 8 , .., 9, 7}. You need to come up with 2 sets, each having 100 elements. You can make them random if you like. 1) Normalize each set in range [0,1]. 2) Find the probabilities of S2 in the ranges [0,0.1], (0.1,0.2], (0.2,0.3], .., (0.9,1] when S1 are in the same ranges [0,0.1], (0.1,0.2], (0.2,0.3], .., (0.9,1] respectively, (find probability of value of S2 in the range, given the value of S1 is in the same range). questions based on chapter 4 exercises: (2) based on exercise 12 Let X be a binomial random variable with mean N * p and variance N * p * (1−p). A) Show that the ratio X/N also has a binomial distribution B) Show that the ratio X/N also has mean p C) Show that the ratio X/N also has variance p * (1 − p) / N questions based on chapter 5 exercises: (3) based on exercise 6 A) Suppose the fraction of undergraduate students who smoke is 15% and the fraction of graduate students who smoke is 23%. If one-fifth of the college students are graduate students and the rest are undergraduates, what is the probability that a student who smokes is a graduate student? B) Given the information in part (a), is a randomly chosen college student more likely to be a graduate or undergraduate student? C) Repeat part (b) assuming that the student is a smoker. D) Suppose 30% of the graduate students live in a dorm but only l0% of the undergraduate students live in a dorm. If a student smokes and lives in the dorm, is he or she more likely to be a graduate or undergraduate student? You can assume independence between students who live in a dorm and those who smoke.