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problem_set1

# problem_set1 - P ABC = P A | BC P B | C P C 8 A B and C are...

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EE 511 Problem Set 1 Due on 27 Aug 2007 1. From the axioms of probability, derive for any event A , (i) P ( A ) = 1 - P ( A ), (ii) P ( A ) 1, (iii) P ( φ ) = 0. A collection { A i } n i =1 is a partition of the sample space S . (iv) Prove for any event B , P ( B ) = n i =1 P ( BA i ). 2. If A B , show that P ( A ) P ( B ). 3. Prove the following identity: P ( A + B + C ) = P ( A ) + P ( B ) + P ( C ) - P ( AB ) - P ( AC ) - P ( BC ) + P ( ABC ). 4. For arbitrary events { A i } n i =1 , prove P ( n i =1 A i ) n i =1 P ( A i ). 5. If P ( A ) 1 - δ and P ( B ) 1 - δ , prove that P ( AB ) 1 - 2 δ , i.e., if A and B are events with probability nearly one, so is AB . 6. If A B , P ( A ) = 1 / 4 and P ( B ) = 1 / 3, find P ( B | A ) and P ( A | B ). 7. A , B and C are three events. Show that P ( AB | C ) = P ( A | BC ) P ( B | C ) and
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Unformatted text preview: P ( ABC ) = P ( A | BC ) P ( B | C ) P ( C ). 8. A , B and C are three events such that (i) P ( A ) > P ( B ) > P ( C ) > 0, (ii) A and B partition the sample space S and (iii) A and C are independent. Can B and C be disjoint ? 9. If two events A and B are independent, show that (i) A and B are independent and (ii) A and B are independent. 10. If three events A , B , and C are independent, show that the events A and B + C are indepen-dent. 1...
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