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hw12_soln - Probability and Stochastic Processes A Friendly...

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Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Goodman Problem Solutions : Yates and Goodman,8.1.1 8.1.3 8.2.2 8.3.2 8.3.3 and 9.1.2 Problem 8.1.1 Recall that X 1 X 2 X n are independent exponential random variables with mean value μ X 5 so that for x 0, F X x 1 e x 5 . (a) Using Theorem 8.1, σ 2 M n x σ 2 X n . Realizing that σ 2 X 25, we obtain Var M 9 X σ 2 X 9 25 9 (b) P X 1 7 1 P X 1 7 1 F X 7 1 1 e 7 5 e 7 5 0 247 (c) First we express P M 9 X 7 in terms of X 1 X 9 . P M 9 X 7 1 P M 9 X 7 1 P X 1 X 9 63 Now the probability that M 9 X 7 can be approximated using the Central Limit Theorem (CLT). P M 9 X 7 1 P X 1 X 9 63 1 Φ 63 9 μ X 9 σ X 1 Φ 6 5 Consulting with Table 4.1 yields P M 9 X 7 0 1151. Problem 8.1.3 X 1 X 2 X n are independent uniform random variables with mean value μ X 7 and σ 2 X 3 (a) Since X 1 is a uniform random variable, it must have a uniform PDF over an interval a b . From Appendix A, we can look up that μ X a b 2 and that Var X b a 2 12. Hence, given the mean and variance, we obtain the following equations for a and b . b a 2 12 3 a b 2 7 Solving these equations yields a 4 and b 10 from which we can state the distribution of X .
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