Homework7 - P X ≥ 3 and compare it to your answers in(a...

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ST561 Fall 2010 Homework 7 Due: Monday, Nov. 15th, 2010 1. Suppose X is a real valued random variable such that P ( X > 0) = 1 and E ( X ) < . Show that Cov ( X, 1 /X ) < 0 . 2. Let X 1 and X 2 be the sample means of two independent samples of size n from the same population with mean μ and variance σ 2 . Use Chebyshev’s inequality to determine n such that the probability that the sample means differ by more than σ is no greater than 0 . 01. 3. Let X have a Poisson distribution with parameter equal to 1. (a) Use Markov’s inequality to find an upper bound for P ( X 3). (b) Use Chebyshev’s inequality to find an upper bound for P ( X 3). (c) Calculate exactly
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Unformatted text preview: P ( X ≥ 3) and compare it to your answers in (a) and (b). (d) Use Jensen’s inequality to find a lower bound for E ( e X ). (e) Calculate exactly E ( e X ) and compare it with your answer in (d). 4. Let X follows a Geometric distribution with parameter p . Let Y = X 2 + 1. Find the probability mass function of Y . 5. Let Y 1 and Y 2 be independent random variables and both follow the uniform distribu-tion on (0 , 1). Let U = Y 1 + Y 2 . Find the distribution function of U and the probability density function of U . 1...
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  • Fall '08
  • Staff
  • Probability theory, probability density function, upper bound, Probability mass function, Geometric distribution

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