homework1

homework1 - failed Consider an experiment that consists of...

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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. Let the sample space be = (0,1], and 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 , 0 ( = 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). 5. A system is composed of five components, each of which is either working or
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Unformatted text preview: failed. Consider an experiment that consists of observing the status of each component, and let the outcomes of the experiment be given by all vectors ( x 1 , x 2 , x 3 , x 4 , x 5 ), where x i is 1 if component i is working and 0 if component i is failed. a. How many outcomes are in the sample space of this experiment? b. Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components 1, 3, and 5 are all working. Let W be the event that the system will work. Specify all the outcomes in W. c. Let A be the event that components 4 and 5 have both failed. How many outcomes are contained in the event A? d. Write out all outcomes in the event A & W....
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This note was uploaded on 10/12/2011 for the course ECE 600 taught by Professor Staff during the Spring '08 term at Purdue.

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