homework1

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 10/12/2011 for the course ECE 600 taught by Professor Staff during the Spring '08 term at Purdue.

Ask a homework question - tutors are online