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zz-10 - i =1 ≤ ∑ ∞ i =1 P A i 4 Suppose P B> Prove...

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Utah State University ECE 6010 Stochastic Processes Homework # 1 Due Friday September 9. Reading G&S, Chapter 1 Exercises 1. Create a list of all the stochastic processes you can think of that might occur in the real world (not just examples from the textbook). Be creative! 2. We defined a field to be a collection of sets that is closed under copmlementation and finite unions. Show that such a collection is also closed under finite intersections. 3. Using the axioms of probability, prove the following properties of probability: (a) P ( A c ) = 1 - P ( A ) (b) P ( ∅ ) = 0 (c) A ⊂ B ⇒ P ( A ) ≤ P ( B ) (d) P ( A ∪ B ) = P ( A ) + P ( B ) - P ( AB ) (e) A 1 , A 2 , . . . ∈ F ⇒ P ( ∪ ∞
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Unformatted text preview: i =1 ) ≤ ∑ ∞ i =1 P ( A i ) 4. Suppose P ( B ) > . Prove the following properties of conditional probability: (a) P ( A | B ) ≥ . (b) P (Ω | B ) = 1 (c) For A 1 , A 2 , . . . ∈ F with A i A j = ∅ for i 6 = j , P ( ∪ ∞ i =1 A i | B ) = ∑ ∞ i =1 P ( A i | B ) (d) AB = ∅ ⇒ P ( A | B ) = 0 . (e) P ( B | B ) = 1 (f) A ⊂ B ⇒ P ( A | B ) ≥ P ( A ) (g) B ⊂ A ⇒ P ( A | B ) = 1 . 5. Prove the law of total probability. 1 6. Prove Bayes rule 7. Suppose A and B are independent events. Show that A and B c are also independent. 2...
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