# hw8 - answer(6 Let X be a continuous random variable with a...

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Fall 2011 ORIE 3500/5500 Problem Set 8 Imagine this is due Friday October 28. Reading: Read Chapter 13. We will move to Chap 11, 14. x/y=page x in course text, problem y. (1) 192/13.4 a, d. Ignore the back of the book. (2) 192/13.5 (3) 193/13.8 (4) 193/13.10 (5) Suppose that the an investor would like to invest in 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 overall. 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
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Unformatted text preview: answer. (6) Let X be a continuous random variable with a density f X ( x ) = 1 2 e-x + e-2 x , for x > 0. (a) Compute the moment generating function of X . What is the range on which the moment generating function is deﬁned? (b) Use the moment generating function from part ( a ) to compute the mean, the second moment and the variance of X . (7) (a) Derive the moment generating function of a Bernoulli random variable that takes possible values { , 1 } with probabilities p, 1-p = q . (b) Derive the moment generating function of the binomial distribution ( n trials, success probability p ) and use it to derive the mean and the variance of a binomial random variable. (Save work. Use 7a to derive the mgf.) 1...
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