Mathematics > QUESTIONS & ANSWERS > Berkeley - CS 70 HW 12. CS 70 Discrete Mathematics and Probability Theory (All)

Berkeley - CS 70 HW 12. CS 70 Discrete Mathematics and Probability Theory

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CS 70 Discrete Mathematics and Probability Theory Fall 2019 Alistair Sinclair and Yun S. Song HW 13 Note: This homework consists of two parts. The first part (questions 1-6) will be graded and will ... determine your score for this homework. The second part (questions 7-8) 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 Short Answer (a) Let X be uniform on the interval [0,2], and define Y = 2X +1. Find the PDF, CDF, expectation, and variance of Y. (b) Let X and Y have joint distribution f(x, y) = ( cxy+1/4 x ∈ [1,2] and y ∈ [0,2] 0 else Find the constant c. Are X and Y independent? (c) Let X ∼ Exp(3). What is the probability that X ∈ [0,1]? If I define a new random variable Y = bXc, for each k ∈ N, what is the probability that Y = k? Do you recognize this (discrete) distribution? (d) Let Xi ∼ Exp(λi)for i = 1,...,n be mutually independent. It is a (very nice) fact that min(X1,...,Xn) ∼ Exp(µ). Find µ. Solution: (a) Let’s begin with the CDF. It will first be useful to recall that FX(t) = P(X ≤ t) =    0 t ≤ 0 t 2 t ∈ [0,2] 1 t ≥ 1 [Show More]

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