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Unformatted text preview: Utah State University ECE 6010 Stochastic Processes Homework # 1 Solutions 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 complementation and finite unions. Show that such a collection is also closed under finite intersections. Given the field F we know that for A, B ∈ F , we have A ∪ B ∈ F and A c , B c ∈ F . Generalizing, when A i ∈ F , i = 1 , . . . , n , then ∪ n i =1 A i ∈ F . Furthermore, ∪ n i =1 A c i ∈ F , as must be the complementary event, [ ∪ n i =1 A c i ] c Hence, by DeMorgan’s theorem, ∩ n i =1 A i ∈ F . 3. Using the axioms of probability, prove the following properties of probability: (a) P ( A c ) = 1 P ( A ) A c ∩ A = ∅ and A c ∪ A = Ω. P ( A c ∪ A ) = P (Ω) = 1 but also P ( A c ∪ A ) = P ( A c ) + P ( A ) (by additivity), so 1 = P ( A c ) + P ( A )....
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 Spring '08
 Stites,M
 Conditional Probability, Probability, Probability theory, #, Utah State University

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