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hw5 - | X |-| Y | d Find Var(4 X-3 Y Problem 4 Let X be a...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Assignment 5 Problem 1 Suppose that the an investor would like to invest into shares of companies A and B . Initially, the shares of the two companies are equally priced, and the investor has a budget to buy K shares over- all. Suppose that the returns X A and X B of one share of each company have the same mean, but potentially different variances: var( X A ) = σ 2 A and var( X B ) = σ 2 B . How many shares of each company should the investor purchase to minimize the variance of the resulting portfolio? Assume that the covariance Cov( X A , X B ) is also known. Do not worry about obtaining a whole number as your answer. Problem 2 Let X be a discrete random variable with possible values 1 , 2 and 4, and p i = P ( X = i ) , i = 1 , 2 , 4 its pmf. If EX = 2, what values of p 1 , p 2 and p 4 ( a ) maximize and ( b ) minimize Var X ? Problem 3 A continuous random vector ( X, Y ) has a joint pdf given by f X,Y ( x, y ) = 6 x 2 if 0 < x < 1, x - 1 < y < 1 - x 0 otherwise . ( a ) Find Cov( X, Y ). Are X and Y independent? ( b ) Find Cov( | X | , | Y | ). ( c ) Find Cov(3

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Unformatted text preview: | X | ,-| Y | ). ( d ) Find Var(4 X-3 Y ). Problem 4 Let X be a random variable with a pdf f X ( x ) = ± 1 if 0 ≤ x ≤ 1 0 otherwise . Let Y = X 3 . Compute Cov( X,Y ). Problem 5 A fair coin is tossed n times. Every time is shows a head (and only in that case), a fair die is rolled and its result recorded. Let X be the sum of all the results of the rolls of the die. Find the mean and the variance of X . Problem 6 A committee of three persons is to be randomly selected from a group consisting of 2 Republicans, 3 Democrats and four Inde-pendents. Let X denote the number of Republicans and Y the number of Democrats on the committee. 1 ( a ) Find Cov( X,Y ). ( b ) Find Var( X + Y ) ﬁrst by computing the pmf of X + Y and then by using your computations in part ( a ) and the formula for the variance of a sum. Did you get the same answer? Due:October 13, at 4 p.m. 2...
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