Homework
Part I
i. Describe how to map values from the interval [-1,1] to the interval [0,1].
ii. 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.
• Normalize each set in range [0,1].
• 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).
Part II
Let X be a binomial random variable with mean N * p and variance N * p * (1-p).
i. Show that the ratio X/N also has a binomial distribution
ii. Show that the ratio X/N also has mean p
iii. Show that the ratio X/N also has variance p * (1 - p) / N
Part III
i. 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?
ii. Given the information in part (I), is a randomly chosen college student more likely to be a graduate or undergraduate student?
iii. Repeat part (II) assuming that the student is a smoker.
iv. 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.
Format your homework according to the following formatting requirements:
i) The answer should be typed, using Times New Roman font (size 12), double spaced, with one-inch margins on all sides.
ii) The response also includes a cover page containing the title of the homework, the student's name, the course title, and the date. The cover page is not included in the required page length.
iii) Also include a reference page. The Citations and references must follow APA format. The reference page is not included in the required page length.