e1q - 1 A , X ( 1 ) = 1 , and X ( 2 ) = 2 . For an...

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ECE 514 , Fall 2007 Exam 1: Due 12:30pm in class, September 27, 2007 Name: 75 mins.; Total 50 pts. 1. (12 pts.) a. Consider a sample space Ω . Suppose C 1 and C 2 are collections of subsets of Ω such that C 1 ⊂ C 2 . Show that σ ( C 1 ) σ ( C 2 ) . b. Let C be the collection of all closed intervals on R of the form [ a, b ] where a < b , a, b R . Is σ ( C ) equal to the Borel σ -algebra? Justify your answer fully. 1
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2. (13 pts.) Consider a probability space , A , P ) . Let B ∈ A be a given event such that P ( B ) > 0 . Define the function P B : A → R by P B ( A ) = P ( A | B ) , A ∈ A . Show that , A , P B ) is a probability space. 2
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3. (15 pts.) Suppose that Ω represents some sample space and A Ω is nonempty. Consider the σ -algebra A = {∅ , A, A c , Ω } , and let P be a probability measure defined on it. Let X be a real random variable defined on the probability space , A , P ) . a. Suppose that ω 1 and ω 2 are outcomes such that
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Unformatted text preview: 1 A , X ( 1 ) = 1 , and X ( 2 ) = 2 . For an arbitrary , what possible values can X ( ) take? Justify your answer fully. b. Given the information in part a and that P ( A ) = 0 . 2 , nd the distribution of X (or charac-terize it in terms of a CDF, PDF, or PMF). 3 4. (10 pts.) Consider a discrete real random variable X , with probability generating function G X . a. Use the Markov inequality to derive an upper bound involving G X , analogous to the Chernoff bound, for the tail probability P { X a } , a R . b. Apply part a to obtain a tail-probability bound for the Poisson distribution with parameter , assuming a . 4...
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e1q - 1 A , X ( 1 ) = 1 , and X ( 2 ) = 2 . For an...

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