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Part II: Distribution Analysis 1. Suppose a random variable has the following distribution. Value: -20 -10 0 10 20 Probability: 0.10 0.20 0.25 0.26 0.19 (a) What is the chance the outcome is at least 0? (b) Given that the outcome is positive, what is the chance it is 20? (c) Find the cumulative distribution function (x) (express it as a piecewise function; a graph may help you do this). (d) Find the average cutcome (expected value), and explain or interpret intuitively. (e) Find the standard deviation, and explain how to interpret it. 2. [Joint Distribution] In the following joint probability distribution, x is the average number of ebook reading hours per day, and y is the average number of (paper) book reading hours per day, for a student selected at random on a certain campus: P(x,y) 1 2 3 0 0.02 0.05 0.08 Y 0.20 0.22 0.05 2 0.24 0.13 0.01 (a) Find the chances a student will spend 1 hour reading ebooks and 2 hours reading (paper) books per day. (b) Find the chances a student will spend at least 1 hour reading ebooks and at least 2 hours reading (paper) books per day. (c) Find the marginal distributions. Are these random variables independent? (d) What is the distribution of the number of hours spent reading per day? (The number of hours reading is the sum of ebook reading hours and paper book reading hours.) (e) Given that someone reads ebooks for 2 hours per day, what are the chances they read 3 hours of paper books per day? 3. Suppose a random variable has the following distribution: Value: 0 0.5 1 1.5 2 2.5 Probability: 0.10 0.12 0.25 0.23 0.20 0.10 (a) Find the average outcome (expected value). (b) Find the standard deviation. (c) Find the cumulative distribution function (x) (express it as a piecewise function; a graph may help you do this). (d) What is the chance the outcome is an integer? 4. Joint Distribution] Consider the following joint probability distribution, P(x,y): P(x,y) -1 1 1 0.04 0.06 0.18 X 2 0.09 0.16 0.14 3 0.15 0.10 0.08 (a) What is the average outcome? (Note that this will be a pair of values.) (b) Find the distribution of the product xy