University of California, Berkeley - CS 70FA19hw11sol Fall 2019, HW 11

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CS 70 Discrete Mathematics and Probability Theory Fall 2019 Alistair Sinclair and Yun S. Song HW 11 Note: This homework consists of two parts. The first part (questions 1-4) will be graded and will ... determine your score for this homework. The second part (questions 5-6) will be graded if you submit them, but will not affect your homework score in any way. You are strongly advised to attempt all the questions in the first part. You should attempt the problems in the second part only if you are interested and have time to spare. For each problem, justify all your answers unless otherwise specified. Part 1: Required Problems 1 Probabilistic Bounds A random variable X has variance Var(X) = 9 and expectation E[X] = 2. Furthermore, the value of X is never greater than 10. Given this information, provide either a proof or a counterexample for the following statements. (a) E⇥X2⇤ = 13. (b) P[X = 2] > 0. (c) P[X ! 2] = P[X  2]. (d) P[X  1]  8/9. (e) P[X ! 6]  9/16. Solution: (a) TRUE. Since 9 = Var(X) = E[X2]#E[X]2 = E[X2]#22, we have E[X2] = 9+4 = 13. (b) FALSE. It is not necessary for a random variable to be able to take on its mean as a value. Construct a random variable X that satisfies the conditions in the question but does not take on the value 2. A simple example would be a random variable that takes on 2 values, where P[X = a] = P[X = b] = 1/2, and a 6= b. The expectation must be 2, so we have a/2+b/2 = 2. The variance is 9, so E[X2] = 13 (from Part (a)) and a2/2 + b2/2 = 13. Solving for a and b, we get P[X = #1] = P[X = 5] = 1/2 as a counterexample. [Show More]

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