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# homework1 - = H Show that ± F = ± G = ± H 4 Let A and B...

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C OMER ECE 600 Homework 1 1. Prove DeMorgan’s Laws, i.e., show that (A B) c = A c U B c and (A U B) c = A c B c . 2. Consider a countably infinite partition C 1 , C 2 , C 3 ,…of a sample space . Show that the collection ± ² ³ ´ µ = = N I , C A : A I i i F where N is the set of natural numbers, is a ± -field. 3. In this problem it is shown by example that two or more distinct collections can generate the same ± -field. Consider the sets { } ] 1 , 3 / 2 ( ], 3 / 1 , 0 ( = F { } ] 1 , 3 / 2 ( ], 3 / 2 , 3 / 1 ( ], 3 / 1 , 0 ( = G { } ] 3 / 2 , 0 ( ], 3 / 1 ,
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Unformatted text preview: ( = H Show that ± ( F ) = ± ( G ) = ± ( H ). 4. Let A and B be two events in a probability space. a. Show that P(A U B) = P(A) + P(B) - P(A & B). b. Show that if P(A) = P(B) = 1, then P(A & B) = 1. c. Show that the probability that exactly one of the events A or B occurs is given by P(A) + P(B) - 2P(A & B). d. Show that if A is a subset of B, then P(A) ² P(B)....
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